inboxes and outboxes

Im interested in implementing inboxes and outboxes.  Inbox is also a core
approved user story.

I just wanted to nail down conceptually what they are, because from
esquiring informally, there seem to be mixed opinions.

The concept of inbox and outbox seems to be best defined in activty pump,
so it would be really nice to hear someone from that community offer some
guidance.

We're interested in the SoLiD community in implementing inboxes, and maybe
also outboxes.

Some questions arise such as:

Does bob post to alice's inbox, or does alice, or both?

When retrieving partial records, what filtering aspects should be taken
into account.

How does discovery work etc.

Any insights would be very welcome!