P2P typically means "individuals' personal computers", but it doesn't have to mean that. For example, blogs can be viewed as peer-to-peer, making connections directly between each other using Trackback or similar technologies. I can make a trackback connection to any blog that supports tracback - on any server - I'm not limited to a particular network of sites that it controlled by a single entity.

I can also envision, using the personal computer P2P approach, a "friends caching for friends" scheme, in which willing members of your social network would "cache" your profile and other public (or semi-public) information. So if you go offline, your profile is still available.

Finally, I think that mobile devices will become the dominant "peer nodes" in such a social network. The location-based possibilities are endless. And a mobile device is more likely to be "always on".