It is common to hear that any of the four sharpshooter companies were "independent." Usually that bit of information is passed on in something of a 'sorry about that' tone of voice. So were they independent?
Well, to a modern understanding of the word -- autonomous, or outside the authority of -- they were not. They weren't just some extra men who went along for the ride with their local regiments and showed up in Washington saying "Hey, we're here too."
To understand why the term is used, we need to go back to its military recruiting usage in the mid-19th century.
Specifically, the men for these New York sharpshooter companies were enlisted along with men for regiments from their areas. The meaning of the term is that the sharpshooters were mustered (or sworn) into service along with them, but were organizationally not part of or 'independent from' the local regiment.
There were other plans for the organization of the sharpshooters - a potential regiment of their own.