Latin (and to some extent Greek) used to be the lingua franca during the middle ages. Later on, French became the language of diplomacy and nobility. Everyone that mattered(1) speaks a local variation of said language which should still be understandable by another speaker. For example, Quebecois and French or American and English.
So, you could have such a language that all the PCs speak. They should be able to interact with everyone else. Now, make sure that each PC speaks the language from where they will go adventuring. If not, they will have to find a teacher and learn the language. This does not take that much time. You can learn everyday grammar and vocabulary in about three to six months of (hard) study. This is what I do for all my games.
Well, almost all my games. If the game is set in a bounded location, then only those languages that are around said location will be relevant. If I set a game in 14th century Venice, I do not need to worry about the PCs speaking Japanese. If I set a game in the Crusades, you better believe that everyone will learn Latin, French, and Arabic pretty damned quickly if they want to get anything done.
If you have boogly powers (aka magic or psionics or whatever), then learning languages could be done via it.
As a side note, Middle Earth started as a setting to play with the evolution of different languages yet most characters manage to communicate quiet well -- and were delayed at the gates of Moria because of a translation error!
Philology is just cool. And just because it is hard to implement in game setting should not be a barrier to trying it out provided that it enhances the enjoyment of the game.
(1) Why, yes sire, I do have blue blood... What about Peasants? They don't need to speak to outsiders, they need to work harder and pay taxes.
Languages don't come from stats, ability scores, or skills. They come from race, and possibly from class or background.
Languages
By virtue of your race, your character can speak, read, and write certain languages. (PHB p.17)
From their first mention languages are set out as a racial benefit. Two exceptions arise--Druidic and Thieves' Cant--as class benefits. Later, we see that knowledge of a language may arise from a background (p.125) or from extensive training (p.187). Which gets us to my interpretation:
Languages are part of your deep background.
Language acquisition as a racial feature arises from assumptions about segregated communities; the DMG (pp.20-21) discusses ways in which one might adjust these assumptions and how that might impact language acquisition. (They give the example of a racially-blind kingdom-dependent language system as an alternative.)
Language acquisition as a class or background feature is more-explicitly based on long times spent in the relevant community/diverse settings/life of study. Note the acolyte and sage gain two languages; guild artisan, hermit, noble, outlander each gain one. Alternately, you can pay to train for 250 days and 250 GP. This also constitutes a large amount of time, effort, and investment on the part of the character.
Thus, changing your ability scores, statistics, or skills have no effect on your languages. Because those changes haven't changed your experience, by which you acquired language. (Admittedly, those changes might impact your ability to hear, speak, read and/or write, however.)
Languages aren't a skill. They aren't tied to an ability modifier and your broficiency bonus in that way because they're binary: you can't (RAW) be more- or less-skilled in a language.
Languages aren't pegged to attributes. But they used to be. Originally PCs were guaranteed two languages: common and alignment. INT>10 made it possible to know more languages. In 1e your intelligence capped the number of languages beyond your base (racial/class) languages that you could know. In 2e your intelligence capped the total number of languages that you could know, all the way down to INT=1 capping you at zero languages: "while unable to speak a language, the character can still communicate by grunts and gestures." (PHB1e p.10, PHB2e p.16)
[ed.: I'm working off of retroclone material for the Original cite: if anyone's got a good cite to edit in I'd appreciate it.]
Is it related to personality? Personality isn't really a defined term in D&D, so this gets sticky, fast. For our intents I think it's easy enough to say "no," but recognize that language and personality formation are interwoven in real life in a way our game just isn't trying to simulate.
Best Answer
The language skills of ropers have varied by edition:
The original roper, in the original Monster Manual, had Exceptional intelligence, but no listed ability to use language.
In second edition’s Monster Manual, the roper is much the same. The Habitat/Society and Ecology sections make no mention of language, and the best we get is “Ropers are not social and do not cooperate with each other.”
The 3.5e roper spoke Terran and Undercommon.
In 4e, those changed to Primordial (which replaced the elemental languages) and Deep Speech (which was said to also be known as Undercommon).
In the preview for 5e, “D&D Next,” that Hoard Of The Dragon Queen was written for, ropers could speak, but they lost that ability by the time 5e was properly published.
It is notable that, prior to 5e, the ropers published by TSR could not speak, but the ropers published by Wizards of the Coast could. For whatever reason, with 5e Wizards decided to go back to TSR’s original.
Anyway, clearly, having a roper who is capable of speech is not exactly unreasonable, and could be done in 5e as a houserule. It would make that encounter in Hoard Of The Dragon Queen run more closely to its intent.
As for which languages, in 5e, Primordial returns (and explicitly has Terran as a “dialect”), as do both Deep Speech and Undercommon, now as separate languages. Deep Speech seems to be used more by the monsters of the deep (e.g. beholders), while Undercommon is more for the more-or-less “civilized” races of the underdark (e.g. drow). Previously, these monsters and races had spoken the same language.
So in 5e, Primordial (with a Terran accent) would definitely apply, and either or both of Deep Speech or Undercommon. Ropers seem more like beholders than drow, to me, so I’d probably go with Deep Speech.