When friends call me on Skype, not just on xbox but on ipad too, Skype does not ring... It doesn't say someone is calling. The same if I try to call them back. I get a missed call notification but never come up saying someone is calling. Even if they have a group call going I can not be called, I can be added into the group but it takes at least 20'minutes and a lot of organising for us to get a call working!
I can be sat with Skype open looking at the screen when my friend is calling but there is nothing coming up on screen at all. I think if I call my friend at the same time he is calling me only then does it actually start a call.
I have a similar issue and it requires you to turn on notifiaction. XB1 will take a little longer to notify (popup at the bottom of the screen) you compared to other devices. For example, my phone will notify on an incoming call, then my tablet and lastly the XB1. You should expect someone to wait at least 15-30 seconds. I use Skype often on my XB1 and all my friends are used to waiting for calls to connect since it's that 30 seconds & time for you to respond. If skype isnt already running, it'll take about a min to load and then 30+ seconds to connect so expect on the short end, 2 min.