There is nothing particularly special about that location of the centre of mass. If you were to find yourself at the precise spot that is the centre of mass of the earth-moon system, the only thing unusual that you would notice is that there would be one thousand miles of rock on top of your head. Pluto is only about twice the size of Charon, so if you put Pluto and Charon on the cosmic seesaw you would find that the balance point is a little bit outside Pluto, rather than inside it. Again, there is nothing particularly special going on there. If you were to find yourself at that precise spot, you would only notice that you were very, very cold and could no longer breathe.
Mike Brown