I'm probably going to get stepped all over by people who are much more versed than I am, but anyway. There is archaeological evidence that Roman forts were built much the same way between Arabia and Scotland. Likewise we can see that similar armor was used across the empire in a given general time period, at least during the Principate.

Does this mean that they were trained the same way? We don't have training manuals from the Principate, let alone manuals by Legion. IMS, the one book of doctrine that survives was written in the very late empire.

However, if the Roman Army had a lick of sense, and considering that they conquered virtually the entire Mediterranean world, they most likely did ... then they would have trained Legions differently depending on where they were stationed. Tactics and formations will be different if one is facing a cavalry-heavy army of Parthian cataphracts in the middle east, vs naked barbarian infantry in the forests of northern Europe.

All of the legions, of course were drilled in the same weapons such as Pilum and Gladius, and in-line battle tactics ... but I have to believe that there was a superset of training on top of that to deal with local foes and conditions. If they wanted to be successful, they'd have to.

Geography and the Political situation plays an important role in the difference between the from frontiers. In the east between Rome and Persia there was a an extremely limited number of routes an army of any significant size could used to invade and all action was effectively bottlenecked around Arminia and Eastern Anatolia. Keeping an Army fully supplied in those area's was a non-trivial task and necessitated moving the actual bases of operations closer to the coast / rivers, and therefor closer to population centers. Having a frontier with another large empire also placed limits on how the local armies could be used. Any large mobilization or expeditions near the boarder risked sparking a large and costly war. This situation resulted in a strategy of a dense in-depth based around static defenses and cities.

On the western germanic boarder was the Rhine River, which could double as an easy supply line for the armies based in Gaul allowing the legion to be further away from the major trade centers. Gaul its self was also less urbanized with fewer large cities than the East. The frontier in the west is also much longer than that in the East, with a much more fluid political situation. This would have necessitated, at times, the legions themselves to be far more dispersed when garrisoning, and employed far more frequently to projectRoman Power across the Rhine. For the Principate, this resulted in a very aggressive forward defense.

The two different strategic situations, against two very different types of opponents would have created an institutional, organizational, and experiential differences between the two frontiers.