The discussion of the technical details of this has such a low signal-to-noise ratio, and I think the blame for that lies mostly with companies who try to set themselves up as infrastructure but maintain significant secrecy about how their tech actually works
Beeper Mini was a completely self-contained app that implemented the iMessage protocol on-device and did not use bridges. Its only optional cloud component was a push notification bridge that wasn't actually given the E2E keys (the cloud proxy would receive your messages but not be able to decrypt them, instead it just sends a push to wake up your device which will fetch and decrypt them).