Sorry, yeah I meant "sharedInbox" vs "PublicInbox". My understanding is that the AP client (assuming you aren't using C2S ActivityPub) sends messages to your AP server which then either sends messages to the sharedInbox of the destination AP instance or directly to the inbox at that other AP instance. I realize the visibility of each message is up to the discretion of each AP implementation but as practiced now, messages are often seen by multiple parties before getting to the final destination.

As far as indexers are concerned, I realize it's entirely possible to use indexers with AP. It's wholly unclear to me how AT Protocol is using indexers or scoping any of their objects so I'm not sure it's worth belaboring the point.