In special relativity, there is no absolute reference frame that determines which object is "actually moving" or experiencing time dilation. According to the principle of relativity, the laws of physics should be the same in all inertial reference frames, which are frames of reference moving at constant velocity with respect to each other.
When two objects are moving at 0.999c away from each other, each object can consider itself at rest and view the other object as moving. From each object's perspective, the other object is experiencing time dilation. This is known as the relativity of simultaneity and the relativity of reference frames.
Both objects can claim to be stationary while the other is moving, and both will observe time dilation in the other object's clock. However, due to the symmetry of the situation, each object will see the other object's clock as ticking slower than its own clock. This is a consequence of the Lorentz transformation equations, which describe how time and space coordinates transform between reference frames moving at relativistic speeds.
In special relativity, there is no preferred reference frame or absolute motion, so both objects can be considered as moving relative to each other, and both will experience time dilation from their respective perspectives.