Sounds like you've already done several of the first things I would have suggested - especially re: getting it off the other device.
Perhaps try... disabling notifications altogether. Close the app (totally), from the multi-tasking bar. Reinstall from the Store. Enable desired notifications.
That's as clean a start as I can think of... :)
i recently signed up and installed the chess.com app on my iphone 3G and on my ipad2 but i don't get any push notifications. checked settings and updates on both..
My advice would be to opt for notifications on only one device (totally shut down & restart the app on both devices), and see if that works.
There are a number of support posts in the Apple forums about people who are unable to get consistent notifications on two or more devices (like this), and that's what we've noticed as well. Unfortunately, this aspect is largely out of our hands.
To begin I'd like to say I've had this problem for some time, and was just waiting for the new update to come out (which has done tonight by the way, and a brilliant job on the Live chess it works brilliantly) to see if it would fix the problem. I've not been recieving push notifications for about two months ever since I downloaded the chess.com app onto my girlfriends iPhone which I had taken by mistake. Ever since I did this I only get badges coming up after I've already loaded the app but not before. I deleted the app off my girlfriends iPhone the same day. I've tried deleted the app and reinstalling it on my phone but still no joy. Can anyone with help? It's really bugging me now.