Ok, so I think I understand the basic concept of Chord, but it seems to me that each ring would have to have every file key hashed to it, so that any node in any identifier ring would be able to lookup any file.  How does this happen?  It seems to me that every ring would have to know about every other ring in order to accomplish this which defeats the benefits of Chord.  Is there something I'm missing?

I think you're on the right track.