Skype doesn't show notification for a specific contact
If you don't receive skype-notifications when one specific person writes you, that's mean it is Disabled :) I don't know how, but suddenly I found the same bug in my Skype. A bug - because I didn't disable it.
How to fix it ?!
Yes, you're right - just enable it :D
Buuuut... I haven't "Notification settings" at all :)
Instead I found another solution - you can enable notifications for any contact just by sending
/alertson into chat with him.
Here some more useful commands from official site. For more information, please, look there.
/add [Skype Name] : Adds a contact to the chat. For instance: /add alex_cooper will add that member to the chat. /alertson [text] : Allows you to specify what needs to appear in a chat for you to be notified. For example, /alertson London will only alert you when the word "London" appears in the chat. /alertsoff : Disable message notifications. /get role : Details your role in the chat. /get uri : Creates a URL link that other people can use to join the group chat. /golive : Starts a group call with other participants of the chat. /info : Details number of people in chat and maximum number available. /me [text] : Your name will appear followed by any text you write. For instance, "/me working from home" will cause the phrase "working from home" to appear next to your name in the chat. You can use this to send a message about your activities or status. /topic [text] : Changes the chat topic. /undoedit : Undo the last edit of your message. /whois [Skype Name] : Provides details about a chat member such as current role.