The common understanding is that, setting air resistance aside, all objects dropped to Earth fall at the same rate. This is often demonstrated through the thought experiment of cutting a large object in half, the halves of which clearly can't then fall more slowly just by being sliced in two.
However, I believe the answer is that when two objects fall together, attached or not, they do "fall" faster than an object of less mass alone does. This is because not only does the Earth accelerate the objects toward itself but the objects also accelerate the Earth toward themselves. Considering the formula:
$F_g = G m_1 m_2/d^2$
We can see that the force of gravity is dependent on both the masses, not just that of the more massive object.
Of course in everyday situations, we can for all practical purposes treat objects as falling at the same speed. But I'm hoping not for a discussion of practicality or what's measurable or observable, but what we think is actually happening.
Am I right or wrong?
What really clinched this for me was considering dropping a small Moon-massed object close to the Earth and a small Earth-massed object close to the Earth. This made me realize that falling isn't one object moving toward some fixed frame of reference, but that the Earth is just another object, and falling consists of multiple objects mutually attracting in space.