Visiting Student/Scholar
I think that there are a lot of different variations, between continents, countries, and even individual universities. It is definitely true that pretty much every school in the US will have its own policy (since there are no national policies regarding registration in the US—although there are some rules that apply to financial aid obtained from federal sources). In Europe, things can also vary greatly between countries and schools (although they're trying to standardize things, this is still very much a work in progress).

Your best bet is to contact officials at the international office of whatever institutions you're interested in attending, and asking them directly.