Indeed, there is no differential time dilation in a frame, in which two clocks move with equal velocities, since they dilate at the same magnitude. Two relatively moving observers always share full amount of time dilation (and Lorentz – factor) depending on the chosen frame of reference. Full amount of time dilation (Lorentz - factor) depends on relative velocity. It is very convenient to look at the problem through the prism of the Transverse Doppler Effect.
Transverse Doppler Effect is purely relativistic effect. Relativistic time dilation has been tested many times by measuring the Transverse Doppler Effect.
In a reference frame in which both emitter and receiver move with equal velocities, they will not measure any Transverse Doppler Effect, i.e. no time dilation. Relatively moving observers will not measure any dilation of each
Just below the diagram:
„Since both emitter and receiver have the speed v relative to this system of reference, there is no differential time dilation.“
Perfect experimental demonstration of said above was the Champeney and Moon measurement of the Transverse Doppler effect. They placed emitter and absorber on opposite sides of the rim of centrifuge. Surely, they measured no Transverse Doppler Effect, i.e. no relative time dilation.
Measurement of the Transverse Doppler Effect demonstrates insolvency of Einstein synchrony convention for the both observers (one – way speed of light is isotropic - c) , which leads to absurd – I am slower than you, you are slower than me.
Reference frame is always mutual property. If A is at rest, B moves then or vice versa.
If the observer A is „at rest“ and synchronizes clocks in his frame by Einstein synchrony convention, observer B must synchronize clocks in his frame by Reichenbach (two – way speed of light is isotropic -c , but one - way speeds of light back and forth are anisotropic), taking into account speed of his own laboratory in chosen mutual reference frame. It is equivalent to introduction of universal for the both observers time. They both must use time of one once chosen reference frame to avoid nonsense - I am slower than you, you are slower than me.
If the both choose a frame, in which they move with equal velocities, the both must synchronize clocks by Reichenbach. Then one - way speed of light will be anisotropic in the each frame, though two - way speed of light will be isotropic anyway. This way they will not measure any time dilation, exactly as in the mentioned above case with Transverse Doppler Effect.
If the observers A and B cannot come to any consensus, who is at rest and who is in motion, they may think that each of them is “at rest”. As a result, each of them may think, that his friend is slower and shorter, which is absurd.