Unfortunately a lot of what happens within the RIAA is private to the RIAA so it's hard to know who is getting the money. Since many artist complain that they don't see any money from online sales it could be that the record label absorbs the money and that the RIAA is not at fault.
Given that the RIAA attend court with numerous lawyers I doubt they make that much money from the lawsuits they win (assuming the defendant doesn't have to pay legal fees of course).
It's important to note that most contracts a label will offer, reward the music copyright and ownership to the label, not the artist. Artists have very little grounds to sue for anything by the time the label is done with them.
Unless you are Metallica, Radiohead, Prince or some other huge band that has fulfilled your 5 album label contract, and now have [a]leverage for a new more even-weighted contract or [b]millions to start your own label and produce and promote your own albums, it is likely you will sell your sole and your ownership of your music to sign a contract like the one described above.