You could configure the sip phones to point directly to one another. No pbx needed. Just “call” and ip address. You only need the pbx to translate/authenticate/provide features. Sip invite comes into phone, making phone ring, pickup phone, and rtp goes directly to phone just like the sip invite. so long as there are no translations causing issues in between, so it actually works best on a private direct network between them.
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All devices like android and ios are 100% firewalled, blocking ALL incoming, unsolicited traffic. No way around this without rooting.
So no, it's not possible as you've described where it is strictly peer to peer with no server involved.
But there is some hope. Apps like Nextcloud Talk use a server, but are 100% open source, free, and hostable on a LAN. And matches what you're after feature-wise.
At least Android devices are not firewalled in any way. Even with the latest Android 14 I can run servers like ssh/ftp/ScreenStream locally on the phone.
There's a firewall/NAT on the phone network, but in the local network it's perfectly possible to connect to other phones (unless the local network has client isolation).