The subject behavior will occur if you copy / paste the Windows Messenger executable file to the desktop, instead of creating a shortcut on your desktop.
To resolve this issue:
1. Delete the Windows Messenger icon from your desktop.
2. If you want a desktop icon, use Windows Explorer to browse to the Messenger folder. The default location is C:\Program Files\Messenger.
3. Right-click msmsgs.exe and press Create Shortcut.
4. Drag the shortcut to the desktop.