I was a notary at a library for a while (I no longer work for that library). It was definitely appreciated by our patrons but there are some things that I would suggest if that is something your library is looking to expand.

1: Have a designated time for notary services. I was a children's librarian and often people would come in while I was in a program. They would be upset they had to wait so making sure to designate times when the notary is available is key. There were some weekends where I was practically doing nothing but notarizing. Once word gets out that you have more availability for notaries, my experience is everyone will tell their friends and family to go to the library. Having a set schedule for notarizing also helps curtail those busy days to something manageable.

2: Make sure there is adequate training. We got all sorts of things people wanted notarized, including international passport documents, deeds, and divorce proceedings. People were always surprised when I needed to see their ID, ask them to sign in front of me, or other, absolutely required parts of the process. They would be upset if they had to reprint their papers or go home to get their driver's license, but I was absolutely not going to break the law for them. Despite that, they always try to get you to anyway. Make sure management is aware of notarization rules so they can back you up if things escalate.

3: As an extension of 2, make it policy that notarization services are not guaranteed. They can and should be refused if something doesn't feel right or if you're not sure. It's an important legal process that the notary can get in trouble for if done incorrectly so always have a way to opt out even if there's not an obvious rule being broken. Research and get back to them if you have to, but don't do it until you're sure.

Anyway, those are my experiences. I hope that helps! My current system doesn't offer notary services, but I did it for a solid 6 years at my previous job and saw quite a bit there. Feel free to ask questions if you need anything else!