I have cats, no dogs (the cats would freak), but I'd say always keep everyone under close supervision, no matter how "safe" the situation might be. Dogs (and cats) are prey-driven. In working with parrots, having dogs/cats/predators around with "eat-y eyes" as I call 'em (that ooh, a new toy, look!), can actually stress them out. (We have birdies too...the beardie doesn't know what to make of the talking parrot. LOL!) I suspect with our little guys being prey in the wild that would be a similar situation with stress from thinking they were going to be eaten all the time. Plus there's the germ factor. So always be watchful and careful.
That said, dogs and cats can be taught. My kitties learned when I said "stop!" they were to halt in their tracks because one of the finches got out. But, those were the middle aged and old kitties nearly ten years ago (all have gone beyond the bridge sadly), and with the "war kittens" there's no teaching them. LOL!
You know your dogs best, but I would be cautious and always keep everyone under observation. It can be done, but slowly and carefully, I'm sure.