The central servers don't have to do much, but they need to exist so new nodes on random IPs can find each other. The Internet has no native service discovery bus.
There’s nothing to stop you doing this through broadcast messages on a local network, though I’m not aware of any way that could be done from within a browser. You could extract the SDP and ICE messages from the browser environment and handle that in Node though, if you were using Electron or something.
Based on: https://webrtc.github.io/samples/src/content/peerconnection/...
You will need to know where the other peer lives. Which is why they expect you to have a discovery service. But typing in local IP's should work for internal networks. The hardest part is getting through NAT and firewalls and that's not a problem on your own LAN.