Same here.
I had a stuck "1" message on my live tile this morning. I removed the tile and put it back and the message indicator went away.
Thats when I noticed that:
a) It doesn't update anymore (live tile, lock screen etc.)
b) It doesn't notify me via Toast Pop up, notification center, or vibration of the phone.
c) It only updates once you open the app.
The only update just recently was the Xbox Music App. There have been no OS updates since the last time it was working fine.
I rebooted and the new Live Tile on the Xbox Music App started working but nothing changed on Skype.
Using WP 8.1 Update 1 (8.10.14157.200) Lumia 930 CV
-----------------
UPDATE:
-----------------
Managed to fix it by going to All Settings -> Notifications -> Skype -> Disable
Close everything, then go back to All Settings -> Notifications -> Skype -> Enable
I also went into Battery Saver and disabled background tasks for Skype, closed everything and then went back in and enabled background tasks.
Testing after this immediately resulted in a toast banner, sound & vibration and a live tile icon showing a message.