WebRTC is about direct peer-to-peer connections between web browsers. The real-time stuff refers to the video/audio codecs that come as part of the spec.

Persistence is handled by storing-and-forwarding your profile information to your friends. Privacy is ensured by broadcast encryption. If you want people unconnected to your network to find you, you could publish public profile information on a directory server (this could also be the status server that's needed for bootstrapping a connection if you're logging in from a public computer at the library or wherever).