To explain why the comment below is correct, you are right that the other instance couldn't get your password. They don't need that to create a post/comment that is supposedly sent from an account though. Those aren't cryptographically signed or anything like that. It just says "this user sent this message".
I don't even think there is a way you could do this with some kind of cryptographic signature, unless it basically just sends an encrypted message to the home server and it verifies it, which then we have the same issue we started with of relying on a single server. I guess every message could be sent with a key, which can be disputed later. If the home server is unavailable, the message is sent, but it can later say that it's a forgery when it comes back online. It's not a perfect fix, but it's the only way this could even possibly work I think.