How different observers measure time?

Suppose I have a coordinate system, call it $S$, in which an observer $O$ is not moving, and $O’$ is moving with constant velocity and another coordinate system $S’$ where $O’$ is not moving and $O$ is moving in a constant velocity, they both have a stop watch and measure the time of the movement of $O’$, what time does each measure?

Since in $S$, $O$ is not moving then he measures the time of $O$ by using the Minkowsky metric, when I change to $S’$ then $O’$ is not moving and he measures time by using a different metric, that is the metric I get after I change coordinates using the Lorrentz transformation.

My question is this idea correct, have I misunderstood something? If yes can you explain what is my mistake?