That's admittedly one big and non-technical problem with that idea. A lot of songs are available on YouTube, so one way is to have people find them on YouTube and paste the URL to add a song to the playlist. You could use the player API to synchronize the audio between listeners.

I would go the user-upload route myself though. Just make it non-trivial for users to rip the songs to their hard drives. By the time you get a C&D (if you do), you may have enough traction to start talking to some of the labels about licensing content.