Bruce Wayne has a massive bounty on his head at all times, multi millions of dollars constantly hanging above him when he goes out in public. Thus, he has his own personal guard when going to galas or charity events. Is it a conflict of interest that his personal guard is made up of his children and coworkers? Probably. Does he care? No.
Socialite 1: Is that... Nightwing? In a suit?
Socialite 2: It is! Who is that next to him? Red Robin?
Dick and Tim, with blazers and sunglasses over their normal costumes, trailing after Bruce the whole night, trying not to giggle at the absurdity of watching Bruce 'Batman' Wayne nearly spill the entire punch bowl on himself:
At first, the public is concerned that a billionaire can "buy out" vigilantes, but it doesn't take long for someone to just... ask Nightwing to guard them for a night, and he agrees, not asking for payment. This eventually becomes little kids asking for Robin to 'guard them on their way to school' and folks begging Batman to 'guard their dog for the week' (Since when did The Mission become pet sitting???) (Obviously, he agrees, tho the little yippy pomeranian gets taken to Wayne Manor for the week instead, and Ace gets a temporary new friend)
A hotline gets set up so that anyone who feels unsafe can call a number and have one of the Bat Family members walk them home. It gets used a lot by drunk college students, women, and kids who stayed up past curfew.
Scared little kid, barely able to reach the payphone, using their last quarters: Um... I n-need.. Robin.. p-please.. I'm really scared... its really d-dark... my house is far away...
Damian, who is already in the area, beelining towards where the call came from: