Some objects may not be public, so we'll need to fetch them with a signed request
If we keep track of which domains have sent signed fetches for a user, we could use that to send their profile updates
This doesn't actually solve the private community issue since the post author's instance can't verify access to the community.
Delaying until we have an actual use case
Potentially relevant: https://git.sr.ht/~mariusor/Fediverse-Enhancement-Proposal/tree/master/item/feps/fep-6606.md proposes an extension to allow querying collections, which would allow other servers to check whether a user is a follower of the community.
However, I'm not sure we actually want to reveal the followers of a community, and so far we don't