Skype message sync and notifications - Windows 8
Solution 1:
If you close the desktop app, the Windows 8 app should notify you as expected.
Skype is a bit weird, it does the same with the iOS client. If I receive a message in the iOS app while it is open, the Windows 8 Skype app won't send a notification.
It seems that as soon as one of your connected apps sets the message as 'read' it disables further notifications to other clients and these clients won't download messages until you open them and wait for them to come in.
In short, Skype in general has issues with notifications.