An extension would be cool! I'm currently trying to do something similar, in some sense; I've patched my instance to filter out DB results from public queries so that only my posts and comments are visible (unless I am logged in).

The only thing I’m not sure about yet is if it’s possible - if I create a Post on an instance that’s not my home, who is hosting the data? Do I only send ActivityPub Create Post with the data and the instance then saves it, or do I create the post on my own instance, send an ID, and if someone requests the Post data on the instance I posted to, it will be requested from mine?

I believe it might be possible, but I'm not sure. It seems that the protocol itself is mostly geared for synchronizing data and distributing updates. From my limited understanding, servers follow users or communities on other servers, which inform those servers that updates should be sent to the requesting inbox. These updates are then used to build up a local copy of the remote page. In the case of a remote community, users interact with their local copy and notify the remote community of those changes.

For example, I am viewing a local copy of this post that I received from lemmy.ml, and my reply to your comment will be stored locally. My server will notify lemmy.ml of this comment (including its contents), and lemmy.ml will notify my inbox if anyone interacts with it (because I am a follower).

It seems that at least some of this syncing might not be necessary... a lightweight frontend could rely on the API of each site it connects with to build up the activities it sends. However, this would probably cause some unnecessary traffic, as such a follower would both receive updates and query the API. Also it would probably break some things, such as ap_id (see the multicolored fedilink icon, which points the original copy of the content on my instance).