"BitTorrent" doesn't freely give your IP... there are different clients and some choose to tell you the IPs, but the program must know these IPs in order to communicate with them. Besides which: they IPs are useful on sites where they ban you for a bad share ratio.

Now the MPAA's problem is this: "the service could become a target for copyright lawsuits" means very little. BitTorrent is literally a protocol and you can't sue a protocol. You can sue the distributors of torrents and those who participate in them and the makers of each individual client, but there is absolutely nothing central - no network whatsoever - to take down. It will be interesting to see what these anti-piracy groups try to do about BitTorrent since there really is no quick fix.