I have read a bit about the Matrix protocol and once tried running the Synapse home server and Element front end which is also available as Android/iOS apps.
Matrix is a federated chat protocol in the same way that ActivityPub is a federated social networking protocol. You can host your own Matrix compatible server and chat with people on other servers.
Recently I watched a video about their recent progress in which they showed a chat working peer to peer over bluetooth low energy when devices did not have access to the internet. Writing this, I just watched another P2P demo video where they show off this functionality once again.
This means you can use your own chat server as usual when
- You have an internet connection on your device
- Your device is on the same network as your server
- Your device can find a P2P route to the other users through nearby bluetooth devices running this software allowing your device to route through other devices network and bluetooth connections
Take all of this with a grain of salt as I am not completely sure I understand how the Matrix services work. I hope this helps.