How will I know when someone sends me an instant message in Skype for Mac OS X?
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If you are signed in to Skype and you receive an instant message when the conversation window is not open, a numbered badge will appear next to the contact in the sidebar. The number displayed indicates the number of unread messages in the conversation.
A red numbered badge will also appear on the Skype icon in the dock. This indicates the number of conversations with unread messages.
If you receive a new instant message from participants in a voice or video call, you can see this icon:
Click it to read the instant message.
Learn more about sending instant messages.