I have heard this stated time and time again (even in this subreddit) despite all the studies I can find putting Stalin's death toll at 6 - 9 million and Hitler's at 14 - 20 million. Is there some historical research I am missing?
When did people start to make this claim and when/how did it become sort of "common knowledge"?
The Soviet archives gradually began to be opened only in the late 1980s/early 1990s. Before that the estimates of Stalin's death toll were of necessity based on speculative and witness evidence.
The speculative evidence consisted, for example, of crude interpretations of the demographic data, such as the bookkeeper Kurganov's simplistic calculations of 66 million demographic losses since 1917 (this includes not only deaths, obviously) on which Solzhenitsyn relies in The GULAG Archipelago. When confronted by the demographer Maksudov about the absurdity of such interpretations, Solzhenitsyn replied that since the Soviet govt hides the data, "we" have a right to any guesses. https://scepsis.net/library/id_956.html Hardly a scholarly approach.
So, the sensationally huge numbers (not based on any credible info) were popularized during the Cold War and then repeated uncritically by many, including the hack historians like Norman Davies, who repeated them in his Europe: a History already in the 1990s, ignoring all the archival finds (the book was a hit of course).
The witness evidence consists of a few statements by various former inmates and Party members who claimed the numbers of the repressed that are an order of magnitude larger than the actual ones that we now know. This is hardly surprising given that witness statements about numbers are rarely reliable. That's a well known problem in the field of the Holocaust research where former inmates would often make exaggerated estimates (the numbers of statements about 3-4-5-6 million Auschwitz victims, for example, is very large), sometimes even claiming having had an access to some sort of special knowledge. This only serves to show that testimonies (in general) cannot by themselves serve as a basis for statistics.
This led to a huge overestimation (often by an order of magnitude, as already mentioned) of the GULAG victims, Great Terror victims etc. in quite a few mainstream publications as well as in the popular imagination - all of which got corrected downwards after the archives were opened.
On the estimates of the people whose death Stalin was responsible for one way or another see this comment: https://www.reddit.com/r/AskHistorians/comments/d7zzp0/is_there_a_reliable_figure_for_the_amount_of/f18qjpn