I intentionally down played what I mean by "bright." There is no source on say, Dalton, Fieldstone or Ethical Culture taking poor smart kids on scholarship. However, trust me, it's done frequently through informal channels. If there is some kid in the ghetto or a public school doing high level calculus in the sixth grade, then calls are made. It's not something you can apply for, it's just something that "happens." And nobody ever talks about it.
Stuy and Bronx Science are competitive, but the vast majority of the kids aren't super bright. They are simply highly motivated.
The genius of NYC is that it continues to foster homegrown talent. A ton of money is dumped into the most competitive schools and most of the graduates return to the city after college to start their careers.