Einstein's train experiment with clocks Referring to Einstein's train hit by lightning bolts thought experiment.
What if, in the middle of the train, together with the moving observer there is a stopwatch, with 2 displays.
One display stops counting when the clock is hit by the beam coming from the left, and the other stops with the beam coming from the right.
At the end of the experiment, would the displays report the same time or different times?
It seems to me that in both cases, one of the observers should see the displays increasing at different speeds. If so, how could it be explained, since the clock is the same and the displays have the same momentum?

 A: Based on the left drawing: Clocks on train, Bolts simultaneous on train: the train observer of course sees the 2 bolts at the same time and the clocks tick to the same number. The train station observer see the right flash first (it travels further to reach the center of the left moving train).
So your question is working backwards: how does the platform observer see the 2 clocks agree (which he has to: if they both display 0.02 he sees a big red 0.02 on each--there is no Lorentz transform that makes a clock display change)--when he thinks they should start at different times? This is good question and the resolution to the paradox is as follows:
The 1st problem is how do the clocks start? Note that they start simultaneously with the bolts in the train's reference frame: the bolt and the timer start have a space like separation, so it's an experiment that can't be done.
Nevertheless, it's a thought experiment: suppose the clocks just happen to be started correctly on the train. What happens is the platform observer says the clocks start at the wrong times. He see the following (time ordered):
1) Right lightning bolt
2) Both clocks start from 0.00
3) Left lightning bolt
4) Bolts hit middle, both clocks stop, reading 0.02.
So for the platform observer, neither clock measured the transit time of the lightning flash it was observing.
A: The left case is not equivalent to the right one. If the bolts are simultaneous on the train frame, they CANNOT be simultaneous on the station frame. So you compare two different experiments. Hence it is normal that results are different ! 
A: Are you asking about this gedankenexperiment?
If so, then the whole point of it is to prove that observers in different reference frames will not be able to agree on the time-order of the lightning strikes.  Your fancy clock-with-two-displays is just a schematic representation of an "observer."

At the end of the experiment, would the displays report the same time or different times?

The answer depends on the clock's frame of reference.  A clock on the train will give one answer.  A clock on the ground will give a different answer.
