Well, there would have been a great deal of paperwork involved, at any point -- the Apollo-Soyuz Test Project was first officially proposed in 1970, and it didn't happen until 1975 (granted, putting people into space took a lot of preparation, but a lot of that time was diplomacy).

But in general, on smaller, less visible projects, it would depend on a lot of things, not least of which is the year: "during the Cold War" is a long time, and relations varied widely during that time. Official exchanges were absolutely a thing: the National Academy of Sciences and the Academy of Sciences of the USSR did establish an agreement involving an exchange program in 1959. How easy it might be for a given scientist to participate in an exchange, however, would depending on a number of factors, like how relations were between the USSR and the US at that time, that scientist's field, or the budget.

I'm not sure about the smaller, more practical procedure of arranging such an exchange, but for a larger-scale view of the program, the National Academies Press actually has a book on the subject of US-USSR scientific cooperation: Scientists, Engineers, and Two-Track Diplomacy: A Half-Century of U.S.-Russian Interacademy Cooperation.