The 2A is a right of the people period as the plain language of the BOR's attests. Who do you suppose "the People" are?
It is interesting to compare the First Amendment and the Second. The First starts out with "Congress shall pass no law" - which to simple ol' me, means it was a constraint on Congress, not anyone else.
Whereas the Second ends with "shall not be infringed" and also, as you noted, contains "the right of the people".
Yet the Second hasn't been incorporated by SCOTUS, whereas the First has (but somehow that "Congress shall pass no law" ditty was disregarded when they upheld McCain-Feingold.)
Which means, at the end of the day, that words have no meaning in this day and age.
"The powers they conferred on this government were to be exercised by itself; and the limitations on power, if expressed in general terms, are naturally, and, we think, necessarily, applicable to the government created by the instrument. They are limitations of power granted in the instrument itself; not of distinct governments, framed by different persons and for different purposes.(...)
The third clause, for example, declares, that 'no bill of attainder or ex post facto law shall be passed.' No language can be more general; yet the demonstration is complete, that it applies solely to the government of the United States. (...)
Had congress engaged in the extraordinary occupation of improving the constitutions of the several states, by affording the people additional protection from the exercise of power by their own governments, in matters which concerned themselves alone, they would have declared this purpose in plain and intelligible language." There is no reason to suspect that the Second Amendment is different.