Historically, the History of Science was originally done by scientists or ex-scientists. The early versions of the discipline were largely what we would today call "internalist" in their orientation towards science: they were about reconstructing the internal logic of historical science. So if you wanted to write about, say, the history of Newton's physics, you needed to have the technical and historical skills to understand his work in context. This would not only mean understanding quite a lot about physics as it is understood today, but understanding a lot about how it would have been understood in Newton's today, complete with outdated terminology and mathematical procedures.

So this by itself meant that the people who went into the History of Science were often very different than people who went into traditional History departments — they had different educational backgrounds, their output could look very different (the most "internalist" of the internalist approaches are hard for a layman to distinguish from a scientific or mathematical paper), and the kinds of questions they were asking of the past could so different as to be somewhat incomprehensible (which is to say, a historian of France can basically understand the questions that a historian of China might be asking, as even if they don't know the local details, they are still going to often be questions about politics, economics, wars, what have you; a historian of physics, however, might be asking questions about the development of 18th-century understandings of turbulent flow, and that is going to look a lot more like "science" than it will "history").

Historically, a lot of the organization of the discipline in the USA (it is different elsewhere; the French also were very interested in the history of science, I understand, but I do not know the details) owes to its early professional formation in the early 20th century at Harvard University. There are a lot of details one can get into, but the short version is that the Belgian refugee George Sarton is generally credited with the transformation of the work from "something done by retired scientists" into "a more rigorous form of historical study of science." (The very earliest writing on the history of science was done by scientists, often retired ones, as noted. It coincides very neatly with the professionalization of science in the 19th century; part of professionalizing, one might argue, is suddenly thinking that the history of one's discipline is worth writing down and considering separate from other history.)

Sarton's professional work also included creating the first journal (Isis, no relation to the terrorist group) and the first American professional society (the History of Science Society). In the 1930s, Sarton's efforts were amplified and encouraged (arguably beyond what Sarton himself wanted to do) by Harvard's new president, James B. Conant, a chemist and scientist-administrator who believed that the history of science needed to anchor a non-scientific education in order to make citizens who understood the importance and functioning of science. This led, in time, to an expansion of education and research in the history of science, and, incidentally, to Thomas Kuhn's The Structure of Scientific Revolutions (which was originally written at Conant's request).

Even today at Harvard the History of Science and History Departments are two quite different beasts, housed in entirely separate buildings (the History Department is in Robinson Hall, a ivy-covered brick building dating from 1904; the History of Science Department is inside the Science Center, a modernist architectural atrocity built in the early 1970s, a fair walk away), with History of Science somewhat carefully preserving its autonomy (though there are professors and students who interact with both). My experience there in the early 2000s was that the two were intellectually quite disconnected; students did not routinely attend one another's seminars or colloquia, and grad students frequently had very little overlapping historiographical canons (I was, to be honest, shocked at how little the History grad students knew about epistemological matters, and how crude their understanding of many basic issues relating to science was).

But despite being one of the epicenters of the discipline, Harvard was not then (and is not now) the center of the world (despite what some Harvardians believe...), and there are many other models for how these fields can work. At some universities today, History of Science is just another subdivision of the History Department. At others it is paired with the History of Technology and/or the History of Medicine. At some it is paired with the Philosophy of Science. At others it is more or less hybridized with Science and Technology Studies, which lends for more influence from disciplines like the Anthropology and Sociology of Science. I have not seen any comprehensive survey of the types of History of Science departments in the USA, but anecdotally there are many of them. It is true that they have many options other than just being part of a History Department, though. (I did my undergraduate work in the History of Science at at UC Berkeley, which had an Office for the History of Science and Technology, which gave it some institutional autonomy as a discipline even though it was formally embedded in a massive History Department; all historians of science employed in universities today are aware there are a million different sorts of relationships possible, and there are only a few programs with dedicated History of Science departments, and even in places that do have dedicated departments, like Harvard, you will also find historians of technology, intellectual historians, and sometimes a rogue anthropologist).

The old disciplinary definitions are less rigid today than they were in Sarton's day (or Kuhn's, or Heilbron's, to just name a few subsequent generations); while it is still the case that many historians of science have undergraduate science degrees, and some even have graduate study, there are also many who have undergraduate degrees in fields of history, or fields of philosophy, or other fields altogether. These do change the scholarship (you cannot really write an "internalist" history of physics without a lot of serious physics knowledge, for example), but the scholarship has radically changed since the 1930s and 1940s anyway. The "externalist" scholarship of the 1960s and 1970s (which emphasized the relationship between science and its broader historical context) frequently put less import on showing the kind of fine-grained change of ideas that the "internalist" approach went for. (My research is on the history of government secrecy and science — there are occasionally technical topics that require careful understanding, but it is not an equation-to-equation sort of endeavor.) In general the demarcations between History of Science and other forms of History have decreased considerably, especially if one is in a department where the lines are less heavily drawn as they still are at Harvard.

I apologize for wending and weaving between the historical aspects and the present day; I talk about the latter not to be definitive about things, but just to point out that the initial assumptions in your question can be poked at a bit. The lines are much blurrier today than they were even 30 years ago, which some people in the discipline think are great, and some people think is terrible. (As with all academic matters, there have been blistering articles written about this for several years.)

For more on the development of the History of Science in the USA, see esp.:

• Michael Aaron Dennis, "Historiography of Science: An American Perspective," in John Krige and Dominique Pestre, eds., Science in the Twentieth Century (Amsterdam: Harwood Academic Publishers, 1997), 1-26.

• Joy Harvey, "History of Science, History and Science, and Natural Sciences: Undergraduate Teaching of the History of Science at Harvard, 1938-1970," Isis 90 (1999): 270-294.