Sure, I think the problem with people running i.e. Mastodon on their own machine would be what should happen when people turn off their computer.

I guess there could be some sort of "super node" solution, where a known Mastodon server would just forward you the "what happened while I was asleep" stuff and I guess people could encrypt their messages to you with your advertised public key, so that you don't have to trust the super nodes, but I am not sure about the additional complexity involved in implementing that vs current Mastodon using ActivityPub, nor about the usability challenges involved.