If you have a situation where you get notifications from one user and don’t receive notification from another user on Skype you can try the following fix as suggested by Skype: Check if both of you are running the latest version of Skype. This ideally should fix the problem of Skype notification not being received from a specific user/s.

If Skype no longer informs you that you’ve received a new instant message even when you and your contacts are online at the same time (from one or more of your contacts), it means that the sender uses an outdated version of Skype with an expired security certificate. To solve the problem, ask them to update their Skype application to the latest version.

Reference: https://support.skype.com/en/faq/FA12325/i-ve-suddenly-stopped-receiving-notifications-for-new-instant-messages
Updated Skype to latest version: http://www.skype.com/go/download-skype

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