This question Why is my internal IP address (private) visible from the Internet? highlights that the Local IP can be accessible from a website because WebRTC needs it.
However it does not answer why WebRTC needs it. From the wiki page of the STUN Protocol :
When a client has evaluated its external address, it can use this as a candidate for communicating with peers by sharing the external NAT address rather than the private address, which is not reachable from peers on the public network
It says nowhere that the Local IP needs to be disclosed. Indeed if both peer have exchanged the the public IP of their respective NAT with the corresponding port that has been assigned to them, they can now communicate and no Local IP is involved.
Could someone provide a simple explanation or situation where the local IP is legitimately needed ?