What you ask reflects perfectly how counter intuitive relativity may appear to us, but it is nevertheless understandable. The key point is that the present is relative (simultaneity is relative).
The normal misconception comes when thinking of it as if A and B measured time with respect to some universal clock. Then, say when A measures 1 sec B measures 0.5 sec, so it is natural to think that B's clock goes more slowly with respect to this universal clock than A's clock. If two events happen at the same time, they happen at the same time for A, and at the same time for B (although the actual time measured by A or B will be different). But now, how would it be possible that this worked the other way around? It seems inconsistent.
The solution comes throwing away the idea of a universal clock. Now, A's present is not the same as B's.
Suppose A could freeze his time. Then, freezing it at 1 sec of his clock he looks where B is and what B's clock is telling. What A sees is that B is some distance away and that B's clock shows 0.5 sec. This is the picture of A's present when its clock tells 1 sec.
Now, imagine B could also freeze his time. B's present when B's clock shows 0.5 sec is completely different from A's present when A's clock shows 1 sec !!! What B sees when his clock shows 0.5 sec is that A is some distance away and that A's clock shows 0.25 sec. So when B freezes his time at 1 sec he sees A's time to be 0.5 sec. This is the actual resolution, for which you have to break with the idea that the present is absolute.
The principle of Relativity is actually the statement that everything A and B measure must be symmetric (and not only mechanical stuff but every physical experiment). This question is very related to the twin experiment, although in that experiment there is an asymmetry because one of the twins is not an inertial observer all the time, and that is what makes that twin to be old when they meet again.