According to Einstein's theory of relativity, there is no absolute reference frame to determine which object is "actually" moving or experiencing time dilation when two objects are moving away from each other. The principle of relativity states that the laws of physics are the same in all inertial reference frames.
In special relativity, the effects of time dilation depend on the relative motion between two observers. Each observer can consider themselves at rest and view the other observer as moving. From the perspective of each observer, it is the other observer's clock that appears to be running slower due to their relative motion. This is known as the relativity of simultaneity and time dilation.
Therefore, according to Einstein's theory, both observers can claim to be at rest and view the other observer as moving and experiencing time dilation. The concept of "absolute" motion or a preferred reference frame is replaced by the idea that all inertial reference frames are equally valid, and the observed effects depend on the relative motion between them.