I think this is a tiny minority, and in fact foreigners don't even have access to subsidized loans (I know that because I fully paid for my education by myself and with personal loans -- I definitely looked for other kind of loans and grants)
I have a grad degree. Certainly I didn't get any help at all, and I don't know any one in my class who got any kind of NSF grants. I studied with someone who got a grant from his govt employer, but he is American.
For example, in my lab every grad students stipend and tuition fee came out of my advisor's grants. Then again in the sciences, if you want to do any meaningful research you have to do a PhD. I was one of the few who was lucky to avoid doing a postdoc (which in most cases is mandatory to try and get a faculty or industry research position). Note that I'd do that PhD 100 times out of 100. Was a ton of fun.