First off, it's just like relative velocity. When two observers have a relative velocity, each one can say the other one is traveling with respect to themself. But you would never then say that both are traveling away from each other at the same time so that their relative speed was then double, would you?
So to make sense of this, we establish a frame of reference which allows us to assign all the speed to just one of them, whichever one we want.
Secondly, time dilation is always a function of the speed of an observer in our chosen frame of reference, the faster he goes, the more his time dilates. If we assign all the speed to the first observer, then he is the only one experiencing time dilation.
So let's say we chose a frame in which the first observer is moving away from the second one who later turns around and returns to the second one. Isn't it clear that only the first observer was traveling so he is the only one that will have been experiencing time dilation and will end up younger than the second one when they reunite?
But we could have chosen the frame in which the first observer was stationary while the first one was traveling away. In this case, the second one will be the only one experiencing time dilation. But when the first one turns around, he will no longer be stationary in our chosen frame, will he? In fact, he will have to go a lot faster than the second observer who is continuing at the same speed as before because he isn't the one who turned around, correct? And because he is going so much faster in our chosen frame of reference, he will be experiencing much more time dilation which will result in him being younger when they reunite.
Now you also asked about how they can each view the other one's clock as running slower than their own. This effect is called Relativistic Doppler and does not depend on any frame of reference, that is, it doesn't make any difference which frame of reference we have chosen.
So let's go back to the first situation in which we have chosen the frame of reference in which the first observer is traveling away from the second one. Remember that only the first one is experiencing time dilation. What does the second observer see? Of course he will see the first observer's clock running slower than his own but he will see it running even slower than time dilation would dictate because there is an added time for the image of the first observer to travel an ever-increasing distance to him.
Now what does the first observer see of the second observer? Well since his clock is being time dilated, he actually travels farther than he normally would between ticks of his clock which makes the ever-increasing time delay for image of the second observer to reach him. The net effect turns out that both observers see the other ones clock as running slower than their own by exactly the same amount.
Eventually the first observer turns around which causes him to see an immediate increase in the rate of the second observer's clock compared to his own. But the second observer will not see any change in the first observer's clock rate until a long time later when the image of his turn-around finally reaches him over the long distance it has to travel. At this point, they both will see each others clock as running faster than their own by exactly the same amount but since the second observer doesn't see this until a long time after the first one, his view of the first ones clock has much less time on it and this holds true until they finally get together.
