Here's a good question, how many of us have both predator and prey in the same house?
Cats and dogs have been a part of my life for far longer than birds ever have. I grew up with dogs and cats. If they didn't live with me, then they lived at my grandmothers house. For ~13 years of my life, I lived in a place that didn't allow cats or dogs, so if we had any pets, they were at my grandmothers house that lived 1/4 of a mile down the road (if that). Except for the birds, and a hamster that belonged to my sister.
My dog is half akita, and, we think, half pit. A combination of two highly prey driven species. He's a happy go lucky idiot of a dog! He's around 80lbs, give or take 5 lbs, and he's terrified of cats that have a vendetta against dogs! Case in point, one of the cats that lives with me.
Sprite ("black" cat) and Squirt ("white" cat)
And Kodi
Squirt loves to terrorize the dog! If Kodi is sleeping on his bed, Squirt will occasionally lay down at the edge of the bed, and then slowly wiggle his way onto the bed, eventually "forcing" Kodi to get up and relocate, which is when Squirt takes over the bed. Or maybe Squirt is laying on a chair and as Kodi walks by Squirt puts his claws out to snatch his fur/skin. Or maybe the dog isn't doing anything at all in particular, and Squirt decides to rush the dog, claws out! Squirt received his name because he's a pain in the arse! Oh, and he brought home a (dead) california quail when he was 4 months or so old? We don't know if he killed it or not, but it wouldn't surprise me, because he is a hunter!
Sprite on the other hand? Not a hunter. Ya, sure, she likes to chase things, but it's not in her to hunt to kill. Squirt and Sprite are siblings! In our old house, she used to pry up the heater registers, crawl through them, and end up in the birds room. She would freak the hell out of the birds, but she never harmed them - even when the birds were living cage-free! Since the registers didn't have screw holes, we ended up duct taping them down! Tacky, but it fixed the issue of her getting out or into the birds room.
I do trust Kodi and Sprite to a degree. I do not trust them with the birds unsupervised, but both do have some trust with me. Squirt on the other hand? Absolutely none!
However, one needs to keep in mind that predators aren't the only danger. Other birds can also be a danger. I lost a bourke parakeet because she escaped from her cage, got into my cherry headed conures area and lost her upper beak. Unfortunately, it happened on a Sunday and I *could not* locate an emergency avian vet in the area. Years later, and we still don't have an emergency avian vet in the area, and the closest one still remains out of state. I ended up losing the bourke the next day.
My mitred conure is another bird that went through a similar incident as the bourke, only he survived. In his foster home, he bullied around two lilac crowned amazons and possibly a yellow nape hybrid? Figuring he could bully them around, he decided to go after the blind ruby macaw. Charlie got the short end of the stick, and two surgeries later, he's missing half of his lower beak. This results in his beak "scissor-beaking", and because of this, he needs his beak dremeled for the rest of his life. It's a two person job that I do myself, since neither one of us can handle someone else doing it for me!