STUN is merely a well-thought out signaling protocol. The simplest, and most common, route is for each party to reach out to the rendezvous (STUN server), which then responds with the port and IP in the other peer. Each party then uses the same address and port it received, because remember that a typical NAT merely maps a port to a port on one of its managed/internal endpoints. The mapping that was created for the STUN server is just as good for use by the other peer.
You could do all of that with your own protocol, but...
Some NATs will include the remote IP in the "key" of the mapping, or even more restrictive approaches, meaning that STUN will have to fall back to other strategies and eventually give up and use TURN.
These days you'll also find that mDNS and other local broadcasts play a role, so that the internet can be avoided entirely for LAN peers.
