Time dilation and Transverse Doppler Effect – where does energy disappear? Kinematic effects of Special Relativity like time dilation and length 
contraction are well known. 
Article in Wikipedia makes it clear:
https://en.wikipedia.org/wiki/Time_dilation
“In the special theory of relativity, a moving clock is found to be ticking slowly with respect to the observer's clock. If Sam and Abigail are on different trains in near-lightspeed relative motion, Sam measures (by all methods of measurement) clocks on Abigail's train to be running slowly and similarly, Abigail measures clocks on Sam's train to be running slowly.”
One practical way to measure the amount of time dilation is measuring frequency of relatively moving source of radiation at points of closest approach – the Transverse Doppler Effect. That means, if Sam radiates with proper frequency f, Abigail measures frequency $f_s/\gamma$. If Sam measures frequency of Abigail, he measures $f_a/\gamma$. That means, the photon redshifts and redshifts again.
According to this logic, if  Sam and Abigail exchange a photon, the photon will finally vanish. Or will Sam and Abigail will vanish themselves? Where  does the energy disappear? 
Please don't confuse it with relativistic Doppler redshift when emitter and absorber recede.
 A: Taken from the Wikipedia article on transverse Doppler shift:

Sometimes the question arises as to how the transverse Doppler effect can lead to a redshift as seen by the "observer" whilst another observer moving with the emitter would also see a redshift of light sent (perhaps accidentally) from the receiver.
It is essential to understand that the concept "transverse" is not reciprocal. Each participant understands that when the light reaches them transversely as measured in terms of that person's rest frame, the other had emitted the light afterward as measured in the other person's rest frame. In addition, each participant measures the other's frequency as reduced ("time dilation"). These effects combined make the observations fully reciprocal, thus obeying the principle of relativity.

A: 
One of practical ways to measure amount of time dilation is measuring frequency of relatively moving source of radiation at points of closest approach – the Transverse Doppler Effect. That means, if Sam radiates with proper frequency f, Abigail measures frequency /. If Sam measures frequency of Abigail, he measures /. That means, the photon redshifts and redshifts again.

Actually, because of relativistic aberration and the relativity of simultaneity these are completely different photons. When in Abigail’s frame the moment that the transverse photon Sam is emitted is a specific event on Sam’s worldline. When that event is transformed to Sam’s frame it is actually emitted after the point of closest approach. And vice versa. So there is no redshifting and redshifting again, and hence no photon disappearing. 

Please don't confuse it with relativistic Doppler redshift when emitter and absorber recede.

This is not possible since they are the same thing. What is transverse Doppler in one frame is redshift when emitter and absorber recede in the other frame. Since you are looking at this scenario in both directions it is unavoidable. 
A: Let's say Sam has some light trapped in a box. According to Agibail that light in the box has momentum into the direction of motion of the box.
Transverse light according to Agibail is such that it has only transverse momentum.
If Sam opens the lid on the top of the box then according to Sam the out flowing light has no longitudinal momentum. According to Agibail it has longitudinal momentum. (Top is the side that points towards Agibail)
So Agibail must instruct Sam to point the box in the backwards direction. And now we can see where the energy "disappears".
It becomes kinetic energy of Sam, according to Agibail. It becomes kinetic energy of Agibail according to Sam.
