How does it work if both observer and person in the train think that the other has slower time?
If both observers go along forever in their respective IRFs, then each will, forever, see the other’s clock’s time as slower. That’s all there is to it. However I suspect the source of your discomfort is along the lines of what’s known as “the twin paradox”, which we will likely be getting to soon. The “paradox” is this: suppose one twin goes on a long relativistic trip and comes back. Another twin stays home on Earth. If each one sees the other’s clock as slow, symmetrically, will they both be the same age when they meet? If not, which one will be older? The resolution to this “paradox” lies in realizing that the symmetry holds only so long as the two reference frames are always moving at speed with respect to each other; in other words so long as they are inertial frames. In the twin astronaut example, if one twin travels away at some speed and comes back, she must necessarily change reference frames, and to do that she has to accelerate. In the simplest case, she goes off in a
