It is highly complicated, and ever-changing which makes it even worse. Some of the long-established parrot species are actually declining, being caught and sent into the pet trade again. Budgerigars once had a sizable population in Florida, yet they are now all but gone, outcompeted for nest sites by other invasives like starlings. Pythons and alligators go back and forth eating each other, and nobody can really say who has the upper hand. Vagrant birds from the Caribbean confound population statuses such as with White-cheeked Pintails. It's become an absolute mess, and I doubt it will ever be fully solved at this point.
Florida has started banning importation and ownership of many invasive species, mostly with reptiles and fish. As far as I've heard most of the birds are generally coexisting with natives without much problem, barring a few like starlings and the Purple Swamphen. The latter faced eradication attempts by shooting, but after the birds kept appearing and the native Purple Gallinule suffered from misidentification, attempts to eradicate them ceased. They are controlled as best as possible, but they are here to stay.
It doesn't even have to be the exotic pet trade either. Red-eared Slider and American Bullfrog are both natives that were introduced to much of the country outside their core range. Pet trade and eating respectively were the reason they arrived, and they essentially have cemented themselves everywhere. Or look at goldfish. There's numerous populations formed from released individuals, and yet they still are sold all over. (Additionally Goldfish suffer from an ethics issue, most people do not keep them properly at all, and the fish live a stunted, short life.)
There's also hunting, which has seen the introduction of various gamebirds and hoofstock. Interestingly most of the successful gamebirds have assimilated themselves into our ecosystem without much conflict. Ring-necked Pheasant, Gray Partridge, Chukar, and Himalayan Snowcock have really only minimally impacted our ecosystems. The hoofstock on the other hand, such as Nilgai in Texas and Gemsbok in New Mexico are having a significant impact on the environment and eradication is the target.
All together a very difficult situation indeed.