This is often expressed as "Each of the twins thinks his clock is moving faster". However a more precise way to say it would be "Each of the twins thinks his clock is moving faster when observed in his own coordinate system."

The difference is important in that if the travelers understand relativity, they will know that their observation only applies into their own coordinate system. They can also calculate and agree with what the other traveler thinks, so they are not disagreeing.

An analogy can be made with movement. When traveler A looks out of his window and sees distance to B's ship increasing, he might think "I'm moving and he is staying where he is". But B can think exactly the same. And yet, both of them understand that their observations are not in conflict, because movement is always relative. Another example from Wikipedia:

While this seems self-contradictory, a similar oddity occurs in everyday life. If person A sees person B, person B will appear small to them; at the same time, person A will appear small to person B. Being familiar with the effects of perspective, there is no contradiction or paradox in this situation.

Another important part of the different coordinate systems is that there is no direct way to simultaneously measure the clock times when they are not next to each other. Because light speed is the maximum speed of any information, what you see of the other clock is delayed more and more as it goes further away.

However, if traveler B decides to turn his ship around and catch up with A, the situation changes. Traveler B's coordinate system now changes as his speed is changing. This breaks the symmetry. By the time when B catches up with A, both of them will observe that B's clock is lagging behind A's clock.