No guarantee that they all pulse at the "same" time but the spec says that iff all have 4 or more sats they will all receive a PPS pulse within +/1 10's of ns. With the sensitivities and out of band rejection available on modern GPS receivers 4 sat fixes are easy today. If one of the remote sensor nodes only has a lock on one sat the math says the error on the PPS pulse goes up to +/- 100 ns.
When I quoted a resulting timing resolution in the microseconds that assumed 4 or more sat fixes. When one of more node had less than 4 sats I would assign a resolution probably in the neighborhood of 10's of microseconds, I have not worked up the math entirely for that statement so it is a little back of the envelope as even 10's of microseconds is good enough for my current needs of sound millisecond resolution.
Short story long: no they do not all arrive at the "same" time, that you can not guarantee for any system there will always be some uncertainty you just have to make the decision is that level of uncertainty small enough for the measurement being made. That is the same thing that is done anytime you make measurements, we get away in a lot of cases today because the instruments available for small amounts of money are usually orders of magnitude better than what we need so the instrument uncertainty gets swept under the rug.
The PPS signal is used quite a bit as a means of data synching if the sites do not have internet connectivity or do not want to, or cannot, rely on internet connectivity.