We do have preferences in terms of online chess notifications. I would start there. If you go to the home menu, and click on account, its going to be the first option in the second column (Manage Online Chess Settings).
Hopefully that helps :)
Online Chess notifications
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I tried an Online Chess game with a friend. After the first moves, neither of us was notified the other had moved until the end of the game. Is that by design or is there a problem with our settings?