How does time dilation work without a privileged reference frame? As I understand special relativity, light travels at the same speed in all reference frames.
What I fail to understand is why time dilation would occur in one reference frame, but not by an equal amount in all other reference frames, at the same time.
To put it another way, if I am sitting on earth, observing a space ship that is flying at a speed near $c$, I should observe him moving through time slower than me.  However, looking out his window, he should observe me traveling relative to him at a speed near $c$ and should likewise see me moving through time slower than himself.
What am I missing from my understanding of special relativity?
 A: If you are standing up, and your friend is inclined on a tilted incline of slope 45 degrees, and you are both the same height, you would say that your friend is shorter by a factor of .707. But from his tilted point of view, you are also shorter by the same factor. There is no contradiction, and there is no need to invoke an absolute notion of up. This is not confusing, because we know there is such a thing as a rotation.
Similarly in relativity, if something is moving, the graph of its motion in time is a straight line tilted away from the t-axis of a non-moving observer, so it is obvious that the equally spaced clock-ticks on the moving line will not be equally spaced to the stationary observer, and that the effect is symmetric, because from the moving point of view, the stationary frame is tilted.
The relativistic transformations are just (hyperbolic) rotations of space into time. They are no more confusing than rotations (although for me, relativity just served to highlight how counterintuitive rotations really are when you think about them deeply).
This is also explained in these answers: What are the mechanics by which Time Dilation and Length Contraction occur? and Einstein's postulates $\leftrightarrow$ Minkowski space for a Layman
A: I like to think of it this way, say you have a mirror like sphere such as a silicone ball you would find on a bird bath, now you look at the your reflection that's in the middle of the ball (THAT CAN REPRESENT YOUR REFERNCE FRAME) and then you look at the perceived edge/border of the visible side of the reflective surface (THAT CAN REPRESENT THE SPEED OF LIGHT BOUNDARY) now from your point of view, you see a fairly normal reflection, where if another person where standing at an angle from you there reflection would appear distorted (REPRESENTING TIME DILATION) but to that other person, they're reflection would appear normal and yours would look warped to there frame of reference, also; if you spin the ball or change positions from around the sphere, everything always looks the same. This is a 3d representation that can be translated to a space-time scenario. 
A: I know this post is 8 years old, but right now I'm in grad school taking a class on Einstein's theories.  I have expressed to my classmates as well as my professor the EXACT SAME CONCERNS as the OP here, but none of the answers I have received seem to solve my issue.  Highly doubt the OP will see this comment, but if you do I feel your pain man! ... such a difficult thing to grasp.
To clarify, I'm not disputing Einstein, I just want to know why my logic is incorrect.
A: The answers so far don't actually address the question. The question can perhaps be elaborated as follows: "A and B are two people, each in a space-ship. At the point of commencement they share the same inertial frame of reference we shall term AB. A then presses the GO button on her rocket and begins to accelerate away from B. But if there are no privileged frames of reference, then from A's perspective isn't this the same as B accelerating away from A? If time dilation occurs at relativistic speeds (let's assume here that A is accelerating at 0.8c) and if from A's perspective it's B who is moving away, then why does time dilation seem to affect only A and not B equally? If A reverses course at some future point and returns to B's initial frame of reference (we assume B hasn't moved along the original XYZ coordinates and only through the t coordinate) then why has B aged much more than A?"  I struggled with this but the answer seems to be because A underwent acceleration away from the (shared with B) initial frame of reference. Although from A's perspective B appeared to be moving away from her, B didn't accelerate away from the initial inertial reference frame and thus didn't experience the time dilation that A is subject to. When A returns to the initial inertial frame AB (as defined earlier) B is much older than A because of A's acceleration away from that initial reference frame. So while there is no privileged frame of reference, the acceleration away from AB by A means that the relativistic effect of time dilation is experienced by A relative to reference frame AB but not by person B because B did not accelerate away from AB. 
