R^2 is on the bottom. We don't ignore the mass of one object because it's insignificant, that would make the top of that equation 0 and the object wouldn't fall at all.

That nifty gravitational law gives you the force of gravity on an object, not the acceleration. Force also equals mass times the resultant acceleration, right? So Fg1 = m1*A1 = G*M*m1/r^2 and Fg2 = m2*A2 = G*M*m2/r^2. m1 and m2 are present on both sides of those equations, respectively, so they cancel, and you get A1 = G*M/r^2 and A2 = G*M/r^2, which are identical. The mass of an object affects the force of gravity, but when you look at acceleration the mass terms cancel out.