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It does the same for me. iPad3 and latest update Apple TV V3.
Mirrors fine up to when the contact accepts the call, then the TV goes back to the Apple TV menu.
BUT if you do this it starts working again. When the call is accepted and the TV goes back to the Apple TV menu, then bring up the mirroring menu on the iPad AGAIN and start mirroring AGAIN. At that point, you will see your contact on the TV.
Not sure what is going on. Obviously Skype does not correctly support the proper video mode for Apple TV full screen, but why does it drop out of mirroring???
Let us know if this fix works for you.