They still have to prove in court that you acquired them illegally. Presumption of innocence still holds, at least in theory.

It's up to them to prove you did anything wrong, which they won't be able to. The RIAA only targets people who illegally distribute digital media (something they can prove).

Copyright law forbids the act of illegal copying and distribution, it does not forbid having files that are (or may) be result of such copying. RIAA would have to argue that you performed that copying illegally; the mere existence of files isn't enough that - for example, in actual convictions of torrenters the main evidence is logs that prove downloading the stuff, not just having files on your computer.

Possession is illegal only for drugs and kiddie porn - for everything else, only distribution really counts.