I would like to replace Ring with something fully local.

Local ML/face recognition would be a bonus. Ability to sync to a private owned server owned by me would be a bonus.

I'm assuming there are projects out there that would enable this -- does anyone have recommendations?

Frigate NVR tied to a home assistant instance has my phone getting proactive notifications about people, birds, and buses (in their select areas...). It's not the easiest thing to setup, but if you're using ethernet cameras it seems to work very very well. The few POS wyze cameras's I have on the system tend to cause some problems, but I know for a fact it's 100% a combination of a) wifi (no matter how 'quality') b) wyze.

So, yeah. Look into frigate.