The only way the RIAA can get you is if they do a search for one of "their" songs (just like you do) and find you as a source for downloading.

They then browse your computer to see what all you have to offer. If you are making a lot of copyrighted material available for others to download, you may wind up getting a certified letter in the mail.

Allowing the files to be downloaded from your computer by others is not unlawful. You are not obligated to protect "their" songs from other people. However, if you have a large number of "their" songs on your computer, they may feel confident that they can convince a judge that you did not buy all of them and put them on your computer, but instead, downloaded (copied) them illegally.

They can't get you for downloading files from their own source computer, because they would be knowingly making those files available over a P2P network.