"That sounds paradoxical, but the only way to compare clocks is to be in the same reference frame, and to do that one of you has to undergo an acceleration. This breaks the symmetry of the situation, and that person is the one whose time was affected. Their clock will be slow relative to the unaccelerated observer"
Ok, both people can not be in the same reference frame, they each have thier own, so if both people look at person A's reference frame, person B's clock is slower. And if both people look at person B's reference frame, person A's clock is slower. There is a paradox, and I'm not seeing what breaks the symmetry.
I am sorry, there are only two reference frames, I'm still trying to learn this stuff. It sounds to me like you are saying that we should ignore one of the reference frames, ignoring one frame gives one result, and ignoring the other gives a different result. That's what I meant by conflict, that was not the correct word to use. I guess I don't see how one reference frame can be ignored.