It would work for techies like us, but it wouldn't work for the segment that is making Facebook so transcendently popular. MySpace was a "teen" thing, others before it, too, but Facebook is mainstream; my grandma has a profile, my parents have profiles, and such is true of many.
Also mainstream is turning off your computer when you're not using it. Almost every non-techie I know is really interested in having their computer turned off. They don't really care about waiting for it to come back up. Telling them that their profile is only accessible while their computer is on is an instant killer. And what if their internet goes out? They're down for that time too. For people with spotty or wireless connections, saying "your profile will only work when your connection works, like 30% of the time", is not going to be too encouraging.
What if a user's hard drive dies? Do you really expect people like my 86-year-old grandmother to make backups of her social network profile frequently enough for it to be meaningful? Or what if a user just gets a new computer? Are they going to understand how to transfer the profile?
Making users download and run a P2P server 24/7 slows down the computer, increases personal connection bandwidth usage commensurate with one's popularity, and creates a big opening for scammers.
I know Facebook would be much less enjoyable if I had to sit around and wait for the dial-up AOL connections of the old people I know to upload their photos. Or even home broadband connections; in the US, those are usually capped at 60-70kbps upload, which is pretty slow.
How easy would it be to create a page that looks like your network's page and say "We've updated our client! This is a special update that we can't auto-update for these smart-sounding technical reasons, so please download and install it." People would do this immediately because social networks are a major part of their web usage; for some, the only reason they use computer. And your network would be poisoned and the users' profile disabled just that easily.
I just don't see how a P2P social network would gain mainstream appeal. What are your solutions to these problems?