Then my hunch is that it is the network connection or lack of a connection. How to debug a mystery to me. Start with trying a “ping” between machines, any VPNs in the way, DNS not working on one or both.
Try doing a FaceTime with someone else?
I have never had to debug FaceTime and I use it all the time, with people around the world. So I have no more thoughts other than the hunch that the issue is not FaceTime.