I've always loved the idea that the Bats are feared in Gotham for the exact same reasons that the Justice League and the rest of the world deems them to be trustworthy and safe
Batman is respected outside of Gotham because of his staunch refusal to kill even when it would be easier. Gotham criminals are terrified because it's well known that his no-kill rule is his biggest hard limit on what he's willing to do. Pretty much everything else goes and they know it
Nightwing is widely beloved for being the nicest and happiest member of the Bats. His smile and constant playful banter are contagious and always make a situation seem better. In Gotham, the only other people who smile and laugh and have that much fun in combat are Harley Quinn and the Joker
Red Hood is a well-known murderous crime lord, but the League is able to trust him because he's one of the Bats. Gotham loves Red Hood the crime lord because his regulation of drug dealers and constant protection of the weak has done wonders for the lower class of the city. He's terrifying because of his open affiliation with the Bats
Red Robin is well known to be a skilled detective who always has a plan to save the day. In Gotham his meticulous investigations and planning are scary because no one can beat them. It's nearly impossible to outsmart someone who is the smaller more calculatedly vicious version of the Batman himself
When Robin doesn't unsheath his sword or bring it with him to a fight, other heroes are glad to see him get used to less violent methods of combat. If someone from Gotham sees Robin enter a fight without a sword, they're terrified because everyone knows Robin deems his sword to be the most efficient weapon for fighting, and if he's not using it things are going to be far more brutal than normal.
Signal is the only Bat that goes out in the day, and is therefore considered less broody and scary to many non-gothamites. Gotham is scared because he's a Bat that shows up during the day.
Orphan/Black Bat is a very skilled hand to hand combatant that the League knows can handle herself. In Gotham, anyone her height that can still win against someone like Bane in a one on one fight is immediately on the 'do not anger' list.
Spoiler is friendly and gets along pretty well with everyone. She's easy to talk to and work with. In Gotham her friendliness has made more than a few give up important information by accident. They like talking to her despite knowing full well that its a terrible idea. She puts them at ease more than she possibly should because she sounds just like one of them
Gotham always flips everything on its head, and that applies to how and why the Bats are perceived the way they are too
