No, both observers in a time dilation scenario do not see the other one moving through time slower. Time dilation arises from the principle of relativity, which states that the laws of physics are the same for all observers in inertial (non-accelerating) reference frames.
Let's consider an example: Suppose there are two observers, A and B, in relative motion to each other. Observer A remains on Earth, while observer B travels in a spaceship at a high velocity close to the speed of light. According to the theory of relativity, both observers will measure their own clocks to be ticking at a normal rate.
However, when observer A looks at observer B, they will perceive B's clock as ticking slower due to the effects of time dilation caused by B's high velocity relative to A. Conversely, when observer B looks at observer A, they will also perceive A's clock as ticking slower due to the same principle.
This may seem paradoxical, as it appears that each observer sees the other's clock ticking slower. However, the resolution lies in the fact that the observers are not in the same reference frame and are therefore subject to different measurements of time. Each observer's perception of time is valid within their respective reference frames, and there is no contradiction.
To clarify, in the scenario described above, if the observers were to compare their clocks after a period of relative motion, they would indeed find that one clock has registered less time compared to the other. This asymmetry is consistent with the predictions of time dilation in relativity theory.