If two Gyarados arrive on the field at the same time, which ones intimidate which?
More broadly, is there something to learn by looking at which Pokémon switches in first?
Hopefully someone more experienced can correct me if I'm wrong.
If it's turn 1, than both Gyarados' should intimidate each other with the faster Gyarados intimidating first. In the case of a speed tie, it would be random. This should also apply if both switch in after both sides have lost a mon (e.g. explosion takes out both mons).
*edit* Quick edit here. As far as I'm aware the switching in message for turn 1 or same time entry is random and has no indication of speed.
After turn 1, it's going to depend on speed stats.
The faster pokemon will switch out first. The switched in pokemon (Gyarados in this case) will use its ability on the opposing mon which is then switched out (again, for another Gyarados in this case), who will activate its ability on the first switched in mon, (the first Gyarados in this case). This interaction won't tell you anything about either Gyarados' speed, but will tell you the relative speed of the previously switched out pokemon which can be very important to learn (discovering a scarfed mon in this way is valuable for example).
While not directly related to your question, if sand or hail are up, the faster mon (again, Gyarados in this example) will be damaged first.