They need to correlate their coordinate systems. Then they will see they both measured the exact same event.OK, now there's another person passing by this first person at high relative speed - let's say half the speed of light. However, since there's no reference nearby, Person B thinks that he's the one sitting still and Person A is moving. When Person A, with the laser setup, pulses the laser, Person B is passing nearby. Person B happens to have a good stopwatch, and times the pulse of laser light himself.
Now Person B sees that the distance from the laser to the photodetector is six feet, just like Person A sees. However, the light beam in his perspective travels farther than six feet - while it goes the six vertical feet, Person A has also moved three feet sideways, so the light has to travel 6.7 feet. Since the speed of light is constant, Person B thinks the light took 6.7 ns (light travels very close to 1 foot per nanosecond) to make the trip.
Who is right?
If they cannot figure that out, they might explain the difference by using some magic like "time dilation".