|MadSci Network: Physics|
It turns out that the laws of physics are different in rotating reference frames, such as your hypothetical person standing on a rotating planet. In other words, there are experiments that you can do to determine whether you are rotating. (For example, observing the angular rotation of a Foucault pendulum.)
In such rotating reference frames, one would see forces that don't exist in non-rotating (or "inertial") reference frames. The "centrifugal force outwards" you mention is one of these fictitious forces. In a non-rotating reference frame, the inward force of gravity makes the two planets accelerate (change the direction of their velocity vector) and "orbit each other". In the rotating reference frame you describe later, one now has the fictitious centrifugal force exactly canceling the gravitational force, leading to no (apparent) acceleration in this reference frame. (So no, the planets won't collide.) Both reference frames will end up describing exactly the same events.
Another example of a fictitious force due to a rotating reference frame is the so-called "Coriolis effect"; see this video for details.
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