The Independent Clans seems a bit of a misnomer, they are at least in New York very much dependent on the Camarilla, but have also proven their use to them. The Ravnos and Giovanni to such a degree that they have received a special position in the city, by Princely decree.
The Ravnos have a loose clan structure, and several of them were instrumental in driving the Sabbat out of New York. These veterans are of course viewed with greater respect than most of the clan receive. The current advisor to the Prince, and de facto leader of the clan in New York is Prisha Anand
The Giovanni have a more rigid structure, the clan in New York lead by the Consiglieri to the Prince, Joseph "Crazy Joe" Giovanni. He runs the family much like the mobster he was, so the structure follows. Never go against the family.
The Followers of Set have no such position, and keep their business to themselves, but are allowed to reside within Camarilla territory. The snakes are present, but not making any noise, waiting in the grass. Current leader, if they have one, is unknown.
Lastly, the Assamites keep a low profile. If indeed any of them at all reside in New York, they most likely make sure to at least nominally claim Camarilla allegiance. This is because New York is a stronghold of the Tremere, the Chantry of Five Boroughs being renowned throughout Warlock circles. Being an independent Assamite in New York means being very careful. There are no known openly independent Assamites in New York.