The short answer is “almost never,” but there’s more to it than that.
Fish names are funny things. Wherever they went, European colonists called the local fish by the names of fish back home in Europe. So the European perch (genus Perca) gave its name not only to the North American yellow perch, but to over a dozen other fish including the Nile perch and the Chilean percha de boca chica (smallmouth perch). Similarly “bass,” from Middle English bars (also meaning "perch") became the white bass, the black bass, and lots of kinds of “sea bass” including, of course, the “Chilean Seabass.” In this case however, it was not homesick colonials, but seafood marketing gurus (the same folks who turned “Slimeheads” into Orange Roughy) that came up with the name. They evidently thought that neither “Patagonian toothfish,” the species’ official English name, nor any of its Chilean names merluza negra, bacalao austral or bacalao de profundidad (“black hake,” “southern cod,” or “deep sea cod”) would be a plus on US restaurant menus or in supermarket fish cases.
Patagonian toothfish (Dissostichus eleginoides)
Back in Chile restaurant owners were having a similar problem with their fish names. Many up-scale (and some decidedly mediocre) restaurants provide English translations of their menus, giving sometimes useful and sometimes bizarre names to their dishes. The most notorious is probably locos con mayo, Chilean abalone with mayonnaise, sometimes translated as “crazies with May.” One of Chile’s most popular and common fish, the corvina, is called the “corvina drum,” or “Chilean croaker” in English, but neither “drum” nor “croaker“ is likely to attract English speaking tourists. So what to call it?
Corvina (Cilus gilberti)
Why not “Chilean sea bass?”
Of course the reason “why not” is that many English speaking tourists know that “Chilean sea bass,” AKA Patagonian toothfish, is one of those fish on the AVOID list for the ecologically conscious. The resulting confusion prompts lots of questions. And some happy, if mistaken, consumers. One visitor wrote “I was able to buy Chilean Sea Bass [at Santiago’s Mercado Central] which retails at $20 a pound in the U.S for under $5 a pound. Very good stuff.”
I’ve never seen or heard of Patagonian toothfish being served or sold in Santiago or anywhere else in Chile outside of the far south, although there are lots of dishes called “Chilean seabass.” So, if you are concerned (or thrilled) about eating dishes translated as “Chilean sea bass” in Chile, don’t be. Unless you happen to be in Chile’s southern-most city, Punta Arenas, you are probably getting corvina. And if you are in Punta Arenas where several restaurants serve it, it will be called “merluza negra” on the Spanish menu. (But if you do find some in Santiago, let me know.) See comments.
Meanwhile the story of Chilean seabass/Patagonian toothfish is interesting in its own right. They occur throughout the southern oceans in cool temperate and sub Antarctic waters, from the east and west coasts of Patagonia eastwards through all of the sub Antarctic islands, submarine plateaus and seamounts to south of New Zealand, and probably in the far south Pacific between there and Chile as well. But since their habitat is in waters from 300 m to over 2000 m (1000 to 6500 feet) deep, they were unknown to science until F. A. Shmitt described and named them in 1898. And it was not until the 1980s that Chilean fishermen, who had been catching them while fishing for merluza australis/Australis hake, which were becoming scarce, began offering them on the commercial market.
Patagonial toothfish range in blue, Antarctic toothfish range in black
Source: “Chilean Seabass”
Toothfish had also been appearing as “bycatch” in the nets of trawlers fishing the waters around the South Georgian Islands (SE of the Falkland Islands) in the 1970s and by the 1990s a fishery for toothfish had developed there as well, followed by Australian and South African fisheries in the southern Indian ocean, and by New Zealand in the Ross sea off Antarctica.
The Patagonian toothfishery is a 'gold mine' fishery; it is a premium product, especially on the U.S. and Japanese market where it is sold for up to $30 U.S. per kilo, it is not a cheap product. When you can realize a lot of money from what may only be a few weeks of fishing, then the fishery becomes extremely attractive.
And what makes it so valuable? “Chef’s Resources,” a web site whose “purpose is to provide culinary resources for chefs, foodies, and culinarians” says:
Chilean Sea Bass is “a wonderfully flavored fish with a high oil content which keeps it moist during cooking” and “which gives it a rich, moist, tender flavor profile which melts in your mouth. It has white flesh with large, tender flakes.”
Nutritionally it is similar to salmon in calories, but with about 2/3 the protein and 1.4 times the fat. (And it's high in mercury; it's recommend that you eat it no more than 3 times a month.)
Chilean Sea Bass Fillet Photo: Le Maitre d’ & Sommelier
The add accompanying the photo says:
It is a versatile fish which can be cooked with white chocolate or champagne. Ten, 10 oz. filets sell for $175
So what’s the problem?
There are several: Only two of its fisheries have been certified as sustainable, toothfish sold as being from those fisheries may or may not be, much of the toothfish sold in the past was harvested illegally and some continue to be illegally caught today, the fishery has been responsible for the deaths of large numbers of sea birds; and it is very dangerous, having resulted in the deaths and injuries to fishermen and the loss of boats.
Sustainability : The Patagonian toothfish is a large predatory fish that grows up to 2 meters in length, may weigh over 100 kg. and lives up to 50 years. They live from mid depths to near the bottom in cold southern waters. They take 6 to 9 years to grow to 70 to 95 cm in length and to become sexually mature. The Antarctic toothfish (D. mawsoni) which is sometimes caught with and sold as Patagonial toothfish, is slightly smaller and lives further south, but has similar characteristics.
Fisheries scientists have argued that heavy exploitation of slow-growing, low-fecundity deep-sea species is inherently unsustainable. The history of large-scale deep-sea fisheries has been a “boom-and-bust” pattern of rapid development, resource depletion, and very slow recovery. As such species, the two toothfishes would appear to be poor candidates for sustainable large-scale exploitation. Both species of toothfish grow slowly, reach sexual maturity after they reach market size, and live in a fragile ecosystem. All of these factors make them inherently vulnerable to overfishing. It is questionable whether large-scale exploitation of such a species could ever be considered sustainable.
In fact however, two fisheries have now been certified as sustainable; the South Georgia Patagonian toothfish longline fishery and the Ross Sea Toothfish longline fishery are certified as sustainable to the standard of the Marine Stewardship Council (MSC), although the sustainability of the Ross Sea fishery has been challenged. 
Identity: Toothfish from certified sustainable fisheries have the label at right, but all may not be well.
A  study conducted by molecular geneticist Peter Marko of Clemson University in South Carolina and several colleagues found several irregularities. Posing as consumers, they bought single fillets of MSC-certified toothfish from supermarkets in 10 states across the United States. Then they looked at short genetic markers to identify the species. Three of the 36 fish samples turned out to be tuna, greenling, and mackerel, the researchers report online today [22 August 2011] in Current Biology. When they examined DNA markers of the 33 actual toothfish, five of these fish had markers that differed from those of fish caught near South Georgia Island. All told, 22% of the samples appeared to be something other than MSC-certified toothfish. "I was stunned," Marko says. The trouble with selling consumers something other than what they want is that it can erode trust in the MSC brand, says Gulbrandsen, which is intended to generate profits that lessen the environmental impact of fisheries.  [emphasis in original]
Illegal, unregulated and unreported (“IUU”) fishing: Toothfish is very valuable, they are found in remote seas and historically there was little policing of the fishery, thus it is unsurprising that a great deal of illegal fishing went on in the past, and some continues today. From 1996-7 to 1999-2000 an estimated 49% of toothfish came from IUU fishing.
Australia’s legal catch quota is 2,900 mt/year, but, in 2003, Australian enforcement agencies estimated that 2,000 mt per month were fished illegally from Australian waters. TRAFFIC reports that the illegal fishery is dominated by Spanish owned fishing interests which employ vessels registered through “flag-of-convenience” states, such as Panama,Vanuatu and Belize. The Chilean fishing industry is alleged to be heavily involved in the illegal trade in toothfish. A good deal of illegal fishing is reported from the Indian Ocean sector of the subantarctic, including areas around Heard and MacDonald Islands. Ports known to support offloading of illegally-caught toothfish include Walvis Bay, Namibia ; Port Louis, Mauritius; Montevideo Port, Uruguay; and many ports in southern Chile. 
In the last decade policing has improved greatly. For example:
In February 2002, the Australian navy captured two Russian vessels fishing toothfish illegally off Heard Island. Some of the difficulties of toothfish enforcement are revealed in details of the story. To make the captures, armed Australian troops and fisheries officials were lowered from helicopters onto the Russian vessels in hazardous conditions of extreme cold and rough seas. The Australian enforcement personnel met resistance from the Russian crews.
Photos: Uncharted Waters 
Thanks to this policing and public awairness, in recent years the IUU catch of toothfish has been dramatically reduced to around 4% of the total catch, according to COLTO (Coalitition of Legal Toothfish Operators). 
Bycatch: Although some toothfish are caught by trawling, the most common method is called “longlining.”
Photo: Toothfish fishing
In this system an 8-12 kilometer main line is anchored to the ocean floor at both ends with marked radio beacons for later recovery. Thousands of baited hooks attached to the main line hang and float at the appropriate depths for toothfish, 800-2,500 metres below the surface. As these baited hooks are cast from the vessel, albatrosses and other seabirds dive for the bait and swallow it, hook and all and are pulled under the water and drowned. In addition, losses to killer and sperm whales that eat the catch and frequently become tangled in the lines have averaged 5% of the total catch and up to 100% on some occasions. IUU fishermen have reportedly attacked whales with dynamite.
Fortunately a new longline system has been developed and implemented in Chile that is reported to have virtually eliminated deaths of birds and substantially reduced the loss to whales by 2006. The extent to which it has been adopted in other fisheries is unknown. 
Danger to fishermen and rescue expenses: Although the toothfishery is not inherently more dangerous than other fisheries, recent incidents in the Ross Sea by unprepared boats or careless crews have cost many lives and great expense. In December 2010 a South Korean fishing boat capsized in the Ross Sea killing half its 42 man crew, and in December 2011 Sparta, a Russian-flagged vessel that was not ice-strengthened, hit ice that ripped a hole in the ship’s hull and requiring the Royal New Zealand Air Force to drop repair supplies to the crew by plane. Rescue efforts were hampered by heavy sea ice, with help only coming seven days later by the South Korean icebreaker Araon. Fortunately, the entire crew survived the ordeal. Another incident occurred on January 11, 2012, when the Korean fishing vessel Jeong Woo 2 experienced a fire on board. Three crew members died, and several others were injured. 
I’ve never eaten Patagonian toothfish, and although I might try it if I go to where it’s caught by artisanal fishermen (Punta Arenas or Ushuaia in Argentinean Tierra del Fuego),I’m not going to be looking for it in LA or Chicago. First it’s too damn expensive; and second, I’m not interesting in eating fish from fisheries that may or may not be sustainable. There are plenty of other fish in the sea.
 “Corvina Drum is a popular fish in South America, but little information exists on its biology, abundance or fishery. They are found from Peru to Chile and around the Galapagos Islands, and are caught by hook and line, longline and gillnets. The abundance of Corvina Drum is not known but their landings have decreased over the past decade. Management is poor overall for Corvina Drum, and essentially the fishery is unregulated. Hook and line fisheries generally cause little habitat damage.” Blue Ocean Institute, Corvina Drum. On line at http://www.blueocean.org/seafood/seafood-view?spc_id=267
 Toothfish Fact sheet FAO 78—Chilean EEZ below 47°S. Coalition of Legal Toothfish Operators inc. On line at http://www.colto.org/fisheries/chilean-eez/ and Patgonian Toothfish, Center for Quantatitive Fishery Ecology, Norfolk, Virginia, USA on line at http://www.odu.edu/sci/cqfe/Research/Southern%20Ocean/Patagonian%20toothfish/Patagonian%20toothfish.htm
Cascorbi, Alice. 2006 (amended 2011) Chilean Seabass, Seafood Watch Seafood Report, Monterey Bay Aquarium. On line at
 Johnson, Genevieve. Voyage of the Odyssey, Log Transcript. On line at http://www.pbs.org/odyssey/odyssey/20020326_log_transcript.html
 Merriam-Webster defines sustainable as “being a method of harvesting or using a resource so that the resource is not depleted or permanently damaged.”
 Cascorbi 2006. Op. cit.
 The MSC is an independent non-profit organization that has developed an environmental standard for sustainable and well managed fisheries.
 Seafoos source staff. 2010. Ross Sea certification criticized. SeaffodNews Environment & Sustainability. On line at http://www.seafoodsource.com/newsarticledetail.aspx?id=8456
 Stokstad, Erik. 2011. 'Eco-Friendly' Chilean Sea Bass May Not Be So Green. Science Now. On line at http://news.sciencemag.org/sciencenow/2011/08/eco-friendly-chilean-sea-bass-ma.html
 Cascorbi 2006. Op. cit.
 Willock, Anna. 2002 Uncharted Waters: Implementation Issues And Potential Benefits of Listing Toothfish in Appendix II Of Cites. Traffic International. On line at www.traffic.org/species-reports/traffic_species_fish20.pdf
 Toothfish fact sheet: The IUU fishery. COLTO. On line at http://www.colto.org/images/Draft-COLTO-IUU-Fact-Sheet.pdf
 Moreno, C.A., R. Castro, L.J. Mújica and P. Reyes. 2008. Significant conservation benefits obtained from the use of a new fishing gear in the chilean patagonian tothfish fishery. CCAMLR Science, Vol. 15: 79–91; Longline fishing, Grenpeace. On line at http://archive.greenpeace.org/oceans/southernoceans/expedition2000/expedition/longline.html; and Patagonian Toothfish. Wikipedia, The Free Encyclopedia.
 Christian, Clair. 2012. Dying for some fish. National Geographic Daily News. Tuesday, January 24, 2012 on line at http://newswatch.nationalgeographic.com/2012/01/17/dying-for-some-fish/, and Mussen, Deidre. 2012. Danger and death in the south's cruel seas. The Press. Jan. 12,2012. On line at http://www.stuff.co.nz/the-press/news/6255575/Danger-and-death-in-the-souths-cruel-seas