I have to say, I see travel from world to world occasionally in my game, but far more common is traveling vast differences. And when you go far enough from home, Common is a different tongue. "Common" in the main play area is actually "Imperial", while if you far enough away, it might be "Peshan" or some other tongue.How many people have characters bridge to other worlds? This has not happened in any game I played in for like 30+ years to the point that it is a non-starter. This also goes to @the Jester ('s) point of having languages be something in you games.
While it makes sense to have common drift into something not the same as another common from another land/world, most play with common being able to talk with others in the game. A point could be made for other races and their language being solid and maybe from their gods to not change, but then there are worlds with things like 'old elvish' or something.
There is no should, only what you decide in your universe. Does it add any value to complicate things?Should Common, Elvish, Orcish, etc. be the same language across different settings like Greyhawk, Forgotten Realms, and Known World? Or each has its own unintelligible version?
I run Greyhawk, and my interpretation is Greyhawk Common is the official language of the Great Kingdom, derived from Old High Oeridian with Suloise and Flannae loan words. (Since Greyhawk is the or setting, I assume the Lake Geneva crew meant something like that, and the concept may not fit as well in other settings with less of a common history for most of the area.)Language barriers are very much a thing in my game. Nobody ever has Common as their native tongue (i.e. what they grew up speaking), and there's no baked-in requirement that players choose Common as one of their PC's languages; thus situations where two or more members of the same party don't share any languages arise on a fairly regular basis. Needless to say, perhaps, magic items that provide translation are quite sought-after.