It's illegal to keep native birds as pets in the US without a permit, as per the Native & Migratory Bird Treaty. There are a few exceptions... native gamebirds (quail & waterfowl) can be kept in captivity, but only if they're captive bred. (Tho I've heard of cheaters who stole eggs and banded the babies as if they were captive bred!)
Also, NON-native birds are fine to keep as pets, even if they are wild. Starlings, house sparrows and pigeons are all not native to the US. (They were introduced from Britain.) In some areas there are also wild (or feral) parrots and other birds. All of these are completely legal to keep as pets. Starlings make nice pets and can even learn to talk... they are related to mynah birds. But only if you raise them from babies - adults stay wild. Sparrows can also be nice pets, much like finches, and if you hand-raise them they're friendly & cuddly.
Some people keep native birds despite the laws, but there's always a risk of being caught at it, fined and the bird confiscated - sometimes resulting in the bird being euthanized.
I have a couple of "wild" pet birds, their photos are on my photo site I'll link to.
- Lara
Lara's Photographs