As usual, Linux needs a bit of under-the-hood wrenching to get things working. HP is actually doing Linux support commendably compared to its peers (see hplip). However, in this case a bit of extra grease was needed.
Symptoms: Printer is detected automatically. A test page fails to print. CUPS logs say:
hp: io/hpmud/musb.c 2095: Invalid usb_open: Permission denied
Solution: Note that this will potentially mess up any HP scanners you have installed. If the P1109 is your only HP device, no problem. I do not claim this to be the correct, the most appropriate, or the