I found
this thread on the Mac OS X Hints site during a frustration-inspired Google marathon.
The jist of it is that the HP Communications software needed to use the scanner functionality of many HP devices has a debilitating issue with Fast User Switching (FUS) in Panther. If you look in /Library/Printers/hp on your system drive, you'll see a program called HP Communications. This program loads automatically when you log in and runs as a background user process. The problem is that when a second user logs in via FUS, a second "HP Communications" process is initiated under that user. The two processes conflict with each other, and either one or both will no longer work.
The only reliable way that I've found to get around the problem is to switch to an administrator account, open Activity Monitor, kill
all copies of the HP Communications process for all users, switch to the account you want to scan from, and run HP communications manually from the /Library/Printers/hp directory. You may also need to unplug your HP device, wait a few seconds, then plug it back in.
The obvious way to fix this issue would be for HP Communications to run as a root process, provided it doesn't create any security risks.