time dilation is symmetrical between inertial reference frames

You see them slow down. And they also see you slow down. And that resolves the paradox. Unless one of you accelerates. In which case only that one actually slows down, while the acceleration is happening.

Time really does slow down, for both observers moving at a constant velocity away from one another. Alice on train counts to 10, and only 5 seconds pass for Bob at a station. At the same time Bob counts to 10, and 5 seconds pass for Alice. It sounds impossible, which is why it was such a big deal when it was proven. It only doesn't make sense if, ironically, you assume a universal frame of reference (that there's a "now" that Alice and Bob can both compare themselves to that's more true than their "now". There isn't.). Relativity makes it all work out because if Alice tries to send a message or affect Bob in any way, for example wave at Bob, and Bob waves back, the events will make sense from each of their own points of view. As long as you don't use instant portals, and maintain speed of causality C, of course.