I'm going to share a real experience that happened to me yesterday. We have had an HP 4730 laserjet MFP donated to us and it has a couple issues I'm trying to resolve. It has a web interface so I typed the ip on the local network into my browser and visited the event log. I noticed a couple errors and decided to search Google for them.
The results page floored me. There, for pages and pages and pages, were the web interfaces of this same printer from around the world. If I clicked on the links, I would be able to control someone elses printer. I clicked on a couple before I realized what I was seeing. They had no password, no security in place and if I were a malicious person, I could have easily changed random settings on any of those printers. A true hacker would be finding ways beyond the printer into the rest of the network.
I can't think of ANY good reason to have a printer on a public facing domain instead of an internal one, but if it's a necessary thing, at least put a strong password on it so a) Google can't index it, and b) you haven't left your printer and your network wide open!
What are they thinking???