In the US, PhDs, in STEM at least, are fully funded. Either by research or teaching assistanship. This is the same as in Europe and students are frequently exchanged.

You'll only be paying for yourself if you're in art history or something stupid like that (even then you could get a TA) or you haven't demonstrated that you're smart enough to get a PhD.

Edit: it's so weird how this completely factual statement has been so controversial. I actually went through this process.