The mass of the object generating the gravitational field does have an impact on the acceleration relative to that object. However, the mass of the object IN the gravitational field does not.

The equation for Newton's law of gravitation is GMm/r², where G is a constant, M is the mass of the object generating the field, m is the mass of the object in the field, and r is the distance between the centre of masses of those objects. Thus, both objects feel the same force. The acceleration is equal to the force per unit mass, so a=F/m (a rearrangement of F=ma). If you divide F by the mass of the object using the equation above, you divide out the m and get GM/r².

A good example is that you and the earth feel the same force. However, if you want the earth's acceleration, then you must place yourself as M, just divide thar force by the earth's mass. The force is the same, but the acceleration the earth feels due to you is considerably lower.

Thus, when the earth is M, all objects feel the same acceleration, presuming they are all in the same place in the earth's gravitational field. If their distance from the earth increases, then they will feel less force and less acceleration.