I started rehabbing wild birds before I knew I wasn't supposed to. A tiny fledgling mockingbird showed up at my front door, and wouldn't take no for an answer. I raised him and a sibling and released them both. Over the next few years, my wife and I raised many mockingbirds. We also raised and kept a non-releasable (scissor-beak) bluejay. We also rehabbed opossums.
Right now we have a live-in flying squirrel. I can't really say she's a pet. She just showed up one day, discovered that she liked parrot food, and made a career decision to stick around.
IMO, rehabbing wild orphan babies of any kind is one of the most rewarding things you can do. I couldn't get a license because I worked an 8-hour day, but I volunteered to work with a licensed rehabber. That way I could stay legal.
You should know that some bird species, namely pigeons, starlings, sparrows, and Muscovy ducks are not legal to release; they're considered to be "non-indigent species." On the upside, they _CAN_ be legally kept as pets. I've raised quite a few clutches, and given them to folks as pets.
Every Spring, most licensed rehabbers are buried up to their eyeballs in baby birds, so are pressured to release them way too early, In the wild, baby birds will stay with their mother until they're bigger than she is. They need her to show them the ropes, explain polite behavior, warn them of dangers, etc., etc. Any rehabbed bird is living on borrowed time, always more at risk than one raised by his mother. The best we can do is give them all the help we can.
Since I lived in the country, my rehabber friend would often let me take the "releasable" birds to my place to release them. What she didn't know was, I kept them an extra two weeks or so, building up their size & strength, so they could take care of themselves.
I also put out food for them until they learned how to fend for themselves.
In that way, I must have raised and released 100 or more birds. It's always gratifying to see one of your charges doing well in the outside world.
About your original question: I've not had any trouble raising birds in the same house as my parrots. With one exception: one of our pet canaries lit on Sterling the Grey's cage, and Sterling killed it instantly. Sad.
I did have another case where I had rescued a baby pigeon and raised him to adulthood. My intention was to keep him as a pet (since they're not legally releasable), but I brought in another pigeon who turned out to have avian virus. It killed them both ISOB!). But my songbirds were unaffected.