Principle of equivalence Einstein's principle of equivalence states that one can not distinguish between a freely falling nonrotating system in a gravitational field and a uniformly moving system in the absence of a gravitational field.
But the following made me wondering: Assuming I had a very light and a very heavy object (like a moon or something) with me. Would'nt the heavy object accelerate faster in the gravitational system because of the stronger gravitational pull of the object itself
$$\mathbf{F}_{12} =
- G {m_1 m_2 \over {\vert \mathbf{r}_{12} \vert}^2}
\, \mathbf{\hat{r}}_{12}$$
compared to the very light object, and thus allow one to distinguish between the two systems?
Or better question: why would this not work?
 A: If you and then moon were together the earth would pull you both. It would pull the moon harder but since the moon is more massive this would (in the absence of other forces) produce the same acceleration as you undergo.
If you factor in the moon pulling the earth up to it then since you and the moon are together the earth is pulled up to both of you.
If you and the moon were standing on a large line with the earth between you then yes the earth and moon will touch before you and the earth touch and that will becsuse the moon pulls the earth harder than you do.
But it doesn't affect the equivalence principle.  The equivalence principle is a local rule. It applies to small regions of space for small regions of time. It says that the inertial frame is one that is falling towards the earth. And thus you standing on the surface of the earth feel the effect of the surface of the earth accelerating you.
Why does the surface accelerate you? It is compressed and so it pushes you upwards. Just like if you were standing on a spring, which actually happens when you stand on a scale to weigh yourself, the spring is compressed so it pushes on you. That's why when you look around everything acts like you'd expect for an elevator accelerating upwards in deep space.
Locally there is no gravity. Any local effect you think is gravity is just what things look like in a frame that is accelerating.  And all the other effects of gravity, tidal forces and global connections, those are due to curvature of spacetime. 
So locally you feel nothing unless something (like the surface of the earth or the floor) is pushing on you. And if it does you'll feel and notice the effects of you accelerating. But this happens in the example of you and the moon. You don't feel anything until the earth's surface touches you.
A: The gravitational force is indeed bigger if the second object has a bigger mass, but the acceleration towards the main body (for example, the moon) remains the same for test bodies of all masses on its surface (assuming the bodies are small, not too far away from the surface, no air resistance etc.). You may recall this from the simple version of Newton's 2nd law: $F=m\stackrel{¨}{x}$, which yields the acceleration independent of the smaller mass.
