If you’re getting the ‘serial number conflict’ dialog, that doesn’t mean that the Personal Use license is not activated. It means that it thinks Panorama is running on more than one computer at a time. The dialog should tell you what the other computer is that is conflict. Is it saying that the laptop is the computer in conflict?
Certainly if the other computer is shut down, Panorama is confused. You might need to reboot the desktop computer.
FYI, the internet has nothing to do with it. Panorama 6 checks for duplicate serial numbers on the local network. Even if a computer is not connected to the internet, it could still be connected to the local network. However, certainly shutting it down completely, as you say you have done, will disconnect it from the local network. So it sounds like the problem is on the desktop computer side.