How would schools that take in multiple nationalities address the language barrier issue?

Question

Although Hogwarts has an attendance of mostly British, Scottish, and Irish students, Beauxbatons, Durmstrang, Castelobruxo, Uagadou, and possibly Ilvermorny and Mahoutokoro all accept kids from other countries.

How would the kids understand each other? Is there an enchantment that makes everything sound like their native tongue? Or is it possible they don't take lessons together?

In addition to the answer to the question, I am also curious as to whether Ilvermorny and Mahoutokoro accept other nationalities.

Answer 1

I went to an international boarding school in England. We had students from all over the world, enough variety that there was actually a set of buntings in the school hall, which featured the individual flags of each student.

Everyone spoke English as a lingua franca

If they didn't speak it well, they often had tools such as Electronic Translators at their disposal.
I frequently heard students from wildly different nationalities speaking English as a common language even though neither of them spoke or understood it well.

In the Potterverse I imagine it'd be pretty much the same, speak the language of your hosting nation, supplemented with a Polyglot Spell of some kind to handle the difficult bits, or perhaps a spell to translate text on paper, so a student would write the word they knew and use the spell to translate it into the other language.

Answer 2

In addition to the example given by Valorum, there are several other examples that show that there is probably no simple magical way to understand different languages. At the Quidditch World Cup we find:

“Couldn’t do me a brew, I suppose? I’m keeping an eye out for Barty Crouch. My Bulgarian opposite number’s making difficulties, and I can’t understand a word he’s saying. Barty’ll be able to sort it out. He speaks about a hundred and fifty languages.”

“Mr. Crouch?” said Percy, suddenly abandoning his look of poker-stiff disapproval and positively writhing with excitement. “He speaks over two hundred! Mermish and Gobbledegook and Troll ...”

“Anyone can speak Troll,” said Fred dismissively. “All you have to do is point and grunt.”

From this we see that Ludo Bagman and his Bulgarian counterpart, both presumably full-fledged wizards, could not communicate without an interpreter. We also see that it is considered very impressive that Mr. Crouch can speak so many languages.

Later in Goblet of Fire when Bagman meets with the goblins we find:

“Absolute nightmare,” said Bagman to Harry in an undertone, noticing Harry watching the goblins too. “Their English isn’t too good ... it’s like being back with all the Bulgarians at the Quidditch World Cup... but at least they used sign language another human could recognize. This lot keep gabbling in Gobbledegook... and I only know one word of Gobbledegook. Bladvak. It means ‘pickax.’ I don’t like to use it in case they think I’m threatening them.”

When Harry reports this to Ron and Hermione, and they wonder why the goblins would have been looking for Mr. Crouch, Harry says:

“Crouch can speak loads of different languages, though,” said Harry. “Maybe they need an interpreter.”

Of course, Harry might be completely wrong, and Bagman was probably lying, but this most likely again indicates that language translation/interpretation is not something that can be accomplished with a simple spell.

At the Second Task we find out that Dumbledore can speak and understand Mermish, and presumably the other people (including teachers, headmasters, and judges) can't:

Dumbledore was crouching at the water’s edge, deep in conversation with what seemed to be the chief merperson, a particularly wild and ferocious-looking female. He was making the same sort of screechy noises that the merpeople made when they were above water; clearly, Dumbledore could speak Mermish. Finally he straightened up, turned to his fellow judges, and said, “A conference before we give the marks, I think.”

While it is possible that this is different because Mermish is not a human language, it seems once again that languages is a skill that cannot be simply replaced with magic.

However, at the end of Goblet of Fire Dumbledore issues an open invitation for the foreigners to return to Hogwarts:

“Every guest in this Hall,” said Dumbledore, and his eyes lingered upon the Durmstrang students, “will be welcomed back here at any time, should they wish to come. I say to you all, once again — in the light of Lord Voldemort’s return, we are only as strong as we are united, as weak as we are divided. Lord Voldemort’s gift for spreading discord and enmity is very great. We can fight it only by showing an equally strong bond of friendship and trust. Differences of habit and language are nothing at all if our aims are identical and our hearts are open.

The statement Differences of habit and language are nothing at all implies that it would not be too difficult to overcome a language barrier if necessary. While Dumbledore was not necessarily referring to them attending classes at Hogwarts, language is clearly not an insurmountable obstacle.

Thus, if there were students of different languages in the same school, there is probably no simple way to magically enable them to understand other languages, so they would probably either learn the main language of the school, or attend classes in their own language.

Answer 3

Given that even powerful wizards seem incapable of language translation spells or enhanced magical learning of languages, students attending those schools would most likely have to learn to speak a common language the old-fashioned way, by attending regular language classes during their early years. This is similar to what happens at most international schools in real life.

‘Harry Potter, you know,’ he loudly told the Bulgarian Minister, who was wearing splendid robes of black velvet trimmed with gold, and didn’t seem to understand a word of English. ‘Harry Potter … oh, come on now, you know who he is … the boy who survived You-Know-Who … you do know who he is –’

The Bulgarian wizard suddenly spotted Harry’s scar and started gabbling loudly and excitedly, pointing at it.

‘Knew we’d get there in the end,’ said Fudge wearily to Harry. ‘I’m no great shakes at languages, I need Barty Crouch for this sort of thing. Ah, I see his house-elf’s saving him a seat … good job too, these Bulgarian blighters have been trying to cadge all the best places … ah, and here’s Lucius!’

_{^{Harry Potter and the Goblet of Fire}}

Presumably in the case of Beauxbatons this common language would be French (since the majority of student are French) although English is also a distinct possibility given that the foreign exchange students as well as Fleur's mother all seem to speak it to a lesser or greater extent, albeit with comedy 'Allo 'Allo accents

At that moment, a voice said, ‘Excuse me, are you wanting ze bouillabaisse?’
It was the girl from Beauxbatons who had laughed during Dumbledore’s speech. She had finally removed her muffler. A long sheet of silvery blonde hair fell almost to her waist. She had large, deep blue eyes, and very white, even teeth.