It would be done under the authority of "the People" - the same authority through which the US Constitution was enacted (or so the logic goes)
Note that the Constitution does not "grant" a right to arms, but recognizes a pre-existing right. Take away the Constitution, and the right remains.
The point is that threatening to take up arms against the government is an ironic companion to constitutional fervor.
Obviously actually engaging in insurrection against a constitutional government would not be constitutional. Anyone taking arms against the government would presumably believe that that government was not operating within the confines the Constitution provides; the act of insurrection would be seen by them as a remedy to an unconstitutional government and therefore neither party would be subject to it.
edit: I see you've discussed this below. Interesting though that gun owners rising up to protect themselves from the government wouldn't be constitutional (it would be done under the authority of the gun, not under the authority of the Constitution).