What is the physics in a spacetime with only two point masses? Imagine a universe consisting of only two point masses, neither of which are in motion. Would they be drawn towards each other? Does there need to be motion for the curvature of spacetime to affect them?
 A: If "spacetime" is assumed to have curvatures obeying general relativity then gravitational fields can do work, hence carry energy, hence have mass. So physically speaking this is not a universe "consisting of only two point masses" because spacetime is not empty. And yes, they will interact with the gravitational field, which will also interact with itself, according to the Einstein equations, producing changes in spacetime curvature that will pull them together.
If space is assumed an empty shell and they interact via Newtonian action at a distance say there can be no change in curvatures by assumption. But there are issues with whether space is even physically meaningful if all that exists are two points.
A: Yes to the first question, no to the second, but there will be motion, as they are drawn together, if I understand you correctly.
The two point masses (let's assume they are Earth mass, but in theory any amount of mass will do), are drawn towards each other by Newton's law of gravity, that is the force pulling them together is proportional to the product of their masses, divided by the square of the distance between them.
So as they get closer to each other, they will feel a stronger and stronger force.  This force provides the motion. The energy for this motion is gravitational potential energy. if they are very far apart in your two body universe, this will be very small, if they are right up against each other, it will be zero. In between these points the force will vary.
