Methods of operating a first endpoint device (ED) facilitate the making of a peer-to-peer (P2P) connection between the first ED and a second ED despite intervening network address translators (NATs) at both ends. The P2P connection extends across a system compliant with the internet protocol (IP) whose architecture that includes a first network (having at least the first ED and a first NAT), a second network (having at least the second ED and a second NAT) and a third network. The first ED connects to the third network via the first NAT while the second ED connects to the third network via the second NAT. A first such method includes: providing the second ED with the first IP-address/port pair; and providing the second ED with first type-information regarding the type of the first NAT.