The accepted answer is not generally correct. Light and heavy objects do not necessarily fall with the same acceleration. Common sense is both common and sense, except among gravitational physicists.
The rules are:
1. The inertial acceleration of a body is proportional to the mass of the attracting body, and does not depend on its own mass.
2. The relative acceleration of two bodies is proportional to the sum of their masses.
3. The time for a body to fall to the Earth is inversely proportional
to the sum of the mass of the Earth and the mass of the body.
When a body is picked up to a certain height and then dropped, the time to fall to the Earth does not depend on the mass of the object. If you lift a ping-pong ball and then drop it, it will take the same amount of time to fall to the Earth as a bowling ball. Splitting the Earth into two masses does not change the sum of those masses, or the free fall time. Contrary to the other answers, the acceleration with respect to inertial space does depend on the mass of the dropped object. This is because the mass of the Earth is reduced by the amount of mass lifted, so that the total mass remains constant. Counterintuitively, a heavier body experiences less inertial acceleration than a lighter body. In this case, it is the relative acceleration that is independent of the mass of the external body.
Now, when an external body is brought to a certain height above the Earth and then dropped, the free fall time does depend on the mass of the external body, because the sum of the Earth and the body obviously depends on the mass of the body. The relative acceleration also depends on the mass of the external body. In this case, the inertial acceleration of the external body is independent of its mass, as is so often claimed.
The third scenario is when two bodies are dropped simultaneously. They will always reach the ground simultaneously. They will experience the same inertial acceleration, which does not depend on their masses.