No, I don't think they should be allowed to just carry a gun just because they bought one. However, if they want to go through training to carry one on duty (like any armed guard or security guard) then absolutely they can.
Handguns are really not good weapons. (They are weapons of convenience. It's way easier to carry a handgun for duty than a rifle... even though the rifle is considerably more accurate due to the longer barrel making it less prone to sway.) Their effective combat range is about 12 yards (36 feet), anything over that and the slightest twitch will send the bullet quite a bit off target. And in high stress situations higher level functions shut down. (Once your heart rate hits roughly 175 you've lost virtually all your motor skills and are relying on pure animal instinct. At 200+ you're in full panic and likely won't even remember what you did for hours or even days later.) So you couple lack of training with high stress and it means a greater danger to everyone.
What I mean by that is in
New York City (2005) around 18% of bullets fired hit their target. That means 82% missed the intended target. (In California, 40% hit but do so at less than 6 feet... which if they are that close when the attack begins, you've been stabbed or injured already.) And that comes from
police statistics... a group of people who are trained to shoot in high stress. Civilians under similar circumstances
will not fair as well. But these bullets that miss still have to go somewhere. They don't stop being deadly just because they didn't hit their intended targets. No, they go on and hit other people or things. (People may not realise this but bullets do go through things like walls and can still be deadly.) And any bullet you fire you are still responsible for. So if you shoot at an attacker and hit another person... guess what? You can be held criminally or civilly liable for injuring that person.
Finally, when emergency personnel do show up they're going to see you with a gun out. And given that they don't know you or what you're doing there, what do you think will happen? (Remember they only know there's a person with a gun inside...)