An object accelerates when the net force (i.e., the sum of all forces) acting on it is non zero.
The force between objects that push on each other is called normal force. The normal force acting between the two skaters pushes each away from the other, and the reason why they move away from each other is, there's nothing to stop them. The boy feels the normal force pushing him to the west, and he feels nothing pushing him to the east, so he moves west. The girl feels the normal force pushing her to the east, but there's nothing to stop her from going in that direction, so off she goes.
The skaters fly apart because each skater experiences a non-zero net force.
The normal force between the skaters is caused by the action of their muscles causing stress and strain within their limbs and, by the fact that their hands can not physically pass through each other.
Two forces act between the rock and the Earth: Gravity is described in Newtonian kinematics as a force, which acts at a distance, attracting the rock and the Earth toward each other. The normal force is caused by gravity and by the fact that they rock and the Earth can not physically pass through each other. It exactly equals the force of gravity, but it acts in the opposite direction: It balances the gravitational force.
The net forces acting on the rock and acting the Earth are zero, so the rock and the Earth do not accelerate in any direction relative to each other.