I can't imagine how such a system would work and also be privacy friendly.

The tags are picked up by other people's phones and reported to a central service. The nature of the reported information unavoidably gives private data like your location to the owner of the service. You could technically build your own network, but you'd have limited users so it wouldn't have the reach of the AirTag network.

Find my Phone software could work in a self-hosted/privacy friendly environment. You'd just need an app that reports its location to a server that you're running. Something like this: https://github.com/LINKIWI/orion-server which seems to use MQTT to periodically report the phone's position information to your own server.