The standard listening port number for a STUN server is 3478 for UDP and TCP, and 5349 for TLS.
You’re really just using a different signaling server here — the IM chat or whatever else you’re using to transfer the SDP offers and answers. Sure, you could theoretically jot those down on a piece of paper and walk them to the person you’re trying to communicate with, in which case your body would be the signaling server
(A STUN server is still used to find out your external IP for NAT-busting. Leaking your IP address to a public STUN server is a privacy dealbreaker there.
Would it be possible to eliminate the need for a STUN server by using IPv6?)