I was experiencing the very same issue. Reinstalling fixed it for me.
Tried a reinstall, deleting temp files and IE history and temp, rebooted and still get the issue.
Edit: Not sure but I may have found the issue. I disabled tweet notification in the app itself and the next notification in the action center wasnt duplicated. I'm guessing the duplication is due to the new integration in 8.1 pulling the notifications directly and turning it on in the app causes the same notification to be displayed twice.
It just occured to me to try this as I remembered I did get notifications after reinstalling the app but before actually signing in which I thought was odd but seems to back my theory.
Edit 2: Confirmed disabling the in app notifications fixed it, toggling it on and off shows its the source of the duplicates. Sorted!