ntfy [1] would work for you entirely in the LAN, if you self-host the server and the phones are connected to the same Wifi. It'll only work for Android phones though, since iOS forces APNS for push notifications.
ntfy is also a distributor for UnifiedPush, so you're not entirely off topic here. Hehe.
Interesting thanks. Have you looked into Apple's Local Push Connectivity API? It's specifically designed for VoIP calls and text messages, but perhaps the latter is enough. Not sure if it requires you to bake in any keys at compile time, or if the iOS app could allow the end-user to configure the push server.
ntfy is also a distributor for UnifiedPush, so you're not entirely off topic here. Hehe.
Disclaimer: I am the maintainer of ntfy.
[1] https://github.com/binwiederhier/ntfy