Current Sith lore is ENTIRELY structured on 'fallen Jedi from the past'. The origins of the current Sith Empire lie with the Jedi Exiles, the 13 fallen Jedi banished from the Republic who conquered the Sith people of Korriban and interbred with them some 3300 years before the Battle of Coruscant. Before then, the Sith HAD built a small Empire after the fall of the Rakata, but their sorcerous ways and the backbiting that comes along with following the dark side led it to collapse centuries before the Exiles' arrival.
The Exiles conspired with the Kissai priest class to utterly subjugate the Sith hierarchy, and used the trappings of the Sith religion and sorcerous ways to legitimize themselves as the 'Sith Lords'. Over the next thousand years, the new Sith Empire rose, fusing the advanced Force-powers of the Exiles (who were capable of creating horrific monstrosities like the Leviathan and Sith Wyrm, as well as Jedi arts like lightsaber-fighting, Sense and Control-type Force powers) with the trappings of the old Sith to create the prototype of what we think of as 'Sith culture'.
Even after the Great Hyperspace War, when the 'pureblood Sith' were thought to be utterly annihilated, the SWTOR Sith Empire was founded by scions of this order. During the 1300 years between the 'original' Sith Empire's fall and the Galactic War, the 'True Sith' had contact with former Jedi like Revan and Malak, and had infiltrators like the Ovair family and the Children of the Emperor disguised amongst the Jedi, manipulating the Republic in preparation for the war that SWTOR is based around. I.E. they knew enough about Jedi arts and culture that they could almost perfectly impersonate them and practice their arts, at least during the 'we beat all the Sith and rebuilt our order from six people, time to crawl into our butts and stop going on dates' period of the Jedi Order.
How would they react to fallen Jedi joining? I kinda feel like the reason this is mostly deprecated in in-game storytelling is to hide that they weren't/couldn't allow faction switching in game...don't add too many former Jedi Sith/redeemed Sith Knights, or the players will get upset they can't do it too. The key for me is that it won't be like with the later Sith of the Brotherhood or Darth Bane's Sith Order...fallen Jedi probably won't be treated like prizes, since the Empire is an insular culture predicated on (and, luckily for the good guys, plagued by) nigh-religious deference to the Sith Lords of Korriban...status by birthright and from the specific arena of 'trained within the fold' are the primary determinors of where you are in the hierarchy, only outstripped by utter heroism in service to the Empire (and even that is limited to Sith Lords...how many legendary army/navy/intelligence heroes do you encounter in-game who die cursing the blindness of the Sith?) In other words: you probably won't get a gold star just for being former Master Drink Bloodnow, you've gotta back it up by getting the goods.