Why don’t I see proper time? Say we have my friend blasting off in a spaceship to Planet Physics and I’m sitting here on earth. I think it’s quite obvious that I will measure the proper length between the planets (because I could use a meter ruler and measure it all)  and that he will measure the contracted length (because his length is squished). 
What I don’t understand is why does he measure the proper time of the journey and I don’t. I can appreciate that my friend should measure proper time because “he is stationary relative to both events (launch and touchdown)” but when I sit on earth I think I’m stationary to the launch and touch down too. So, why can I not measure the proper time it takes to reach planet physics? Can you explain why I am not stationary relative to touch down (I always see the planet staying still in the sky!)
 A: The proper time is by definition the time as measured by an inertial observer who is present at both events.  That's your friend and not you.
A: Assuming flat spacetime for simplicity, there is a proper time $\Delta\tau$, associated with two events with time-like interval, related to the Lorentz invariant interval $\Delta s^2 = (c\Delta t)^2 - (\Delta r)^2$:
$$c\Delta\tau = \sqrt{\Delta s^2}$$
This proper time is equal to the elapsed time (between the events) given by an inertial clock with world line through both events. Put another way, an unaccelerated clock co-located with both events, will measure the proper time associated with the events.
But in general, there is the proper time $\tau$ along the world line of an accelerated observer through two events (which will always be less than the proper time associated with the two events). This proper time is defined as the elapsed time between the events given by a clock with a world line through both events.
The clock on the spacecraft records the proper time along the accelerated world line from the launch event to the landing event. While the world line of your clock is through the launch event, it is not through the landing event.
A: As other have pointed out, the concept of "proper time" is what Einstein discarded in special relatively. You each experience time and distance differently but in a mutually consistent way and reconcilable way.
You scenario is however not covered by special relativty. 
If your friend was making a fly-by Earth follwed by a fly-by of Plant Physics having maintained a constant course and speed, (e.g. without accelerating and decelerating) then you would bot be in "inertial frames" with a constant relative velocity between you and him, which is what special relativity covers.
Because you and your friend start and end with no or little relative velocity but he has had undergone high and sustained acceleration on the space ship while you have not, then your experiences are not symmetrical, there is no "paradox" with him having aged less than you and when he sends you a message to say he arrived, you may be long dead of old age.
A: Special relativity is not special. It is simply a process where an observer A measures observer B when they are in relative motion. There is no preferred frame of reference. Thus observer A cannot say they are stationary (preferred frame) while observing B. They both move away from each other. That is what relativity is. 
Both observers see each other’s “clocks” run slower. When they return their clocks are identical. This is the result of using a method of observation with a particle or wave that has a finite speed. 
Special relativity also applies to sound waves. In fact any wave. 
Submarines that use sonar actually observe the other moving submarine shorter than its length when both are stationary or moving in tandem. It is the result of the finite speed of the sound wave. 
Lengths do not physically shrink. The just appear to shrink. It is an optical illusion only. 
