How did you share the printer? There are two methods, one is via samba (windows) and the other is via ipp/cups (unix). As far as I know, cups doesn't require authentication by default unless you've set it up that way. Samba on the other hand does by default.
I would recommend using cups sharing, ipp works in windows clients as well as any other client and is well supported in Ubuntu. You simply go to your
System > Administration > Printing and select
Server > Settings.
You can then visit the ip address of your server in any web browser at port 631 to check you get a connection and access to the printer:
If you do this, your other machine should see this shared printer automatically. Configuration should then be automatic and you should not have any passwords to deal with.