if that last one (GET'ing from someone's outbox) was the only way to see what people have sent, this wouldn't be a very efficient federation protocol!
? So at core, AP would require cycling through people's outboxes to get the stuff that is available there. Such pulling may be enough for just specific circles of people. Would it be enough for dAPplr? Come and get it?
You can POST to someone's inbox to send them a message (server-to-server / federation only... this is federation!) You can GET from your inbox to read your latest messages (client-to-server; this is like reading your social network stream) You can POST to your outbox to send messages to the world (client-to-server) You can GET from someone's outbox to see what messages they've posted (or at least the ones you're authorized to see). (client-to-server and/or server-to-server)
get and post behave differently on the client and server side. Get from inbox client side, is reading ones stream. Post to outbox client side is sending messages out. Post to inbox server side is incoming stuff from the fediverse. Get from outbox serverside is getting the msgs you're authorised to see. This last one would be how I'd approach e.g. dAPplr (I think I found my working title)