This I believe is a permissions issue.
To confirm it's a permissions issue, map a network drive to the desktop PC from the laptop. Then try to print.
If it works (as I suspect it will), then to ensure this continues you have two choices:
1) Continue to keep a mapped drive to that desktop PC (as the credentials used when mapping to that drive will be passed to the PC when a print job is created).
2) Enable the "guest" account on the desktop PC and set the password to NULL (i.e. no characters). This will allow other computers/users outside of the desktop's security realm to log into it when it sends the job.