Usually Batman is fine with the police using them to do their job (although he still disliked the idea of Dick potentially using one.) He has trained his students to learn how to use the weapon for better understanding of what their up against. Jason Todds' time as Robin was interesting in this regard. Death of the Family has flashbacks of his training after his death and sure enough Bruce shows him how to fire a gun.
Now Alfred has a bit of history that varies depending on the canon. Sometimes he's someone that used to work in the field prior to working for the Waynes. Dini had him acting as something of a body guard for Thomas. There may have been an agreement between Thomas, Martha and him that he'd carry protection to use as he saw fit. Bruce may have respected that agreement since it was made with his parents. Whatever the reason it's never questioned when Alfred is seen carrying a shotgun. Booster Gold fans know he's willing to fire it as he shot Rip Hunter (not shown where he was shot although he lived through the ordeal. Then again time travel.)
When Jason returns as the Red Hood the guns are a big deal, their basically a symbol of him picking up the opposite ideology of Bruce. In the relaunch it's not as big of a deal anymore. Of course in these situations it's usually when Bruce wants something from his former sidekick. In Batman and Red Hood #20 he literally holds out the holsters asking Jason to suit up because he's "seeing red." When working together he reminds him not to aim to kill and the request is obeyed. Jason doesn't usually use them to take a life while working with the bats or when he's in Gotham. Is that why Alfred has some leeway? As long as he doesn't kill?
Curiously Jasons' guns haven't been seen in comics for awhile as writers have taken to having him favor his swords. I think this may have to do with Dick taking them up. Regardless I'd like to see this explored more in comics.