Do we ever find out what the language of instruction used at Durmstrang Institute is? At Beauxbatons it would have to be French, but although “Durmstrang” sounds German, the Durmstrang students we meet are all of Slavic ancestry as far as we know.
On the one hand, German continues to be sort of a “lingua franca” in a lot of Eastern Europe; this is a relic of the days when most of the cities and towns were mainly German-speaking, and Slavic, Romanian and Magyar were the dialects of peasants. For a long, long time, up till the end of WWII, a lot of the professional classes and middle-classes in Eastern and Southeastern Europe were German-speaking, and I’m told that to this day, German is a good language to know if you don’t know the local speech in these areas—still more widespread than English.
On the other hand, the only Durmstrang types we get to meet have Slavic-sounding names (although this, in itself, doesn’t necessarily mean much; there was a lot more movement between countries in Eastern Europe than most people are aware of in the U.S.) so the language used may be one of the Slavic tongues. Since many of these languages have high default rates (for example, if you know, say, Polish, you have a big leg up on learning Russian and Czech) they could require that students coming in from backgrounds that do not speak the language of instruction must first master at least enough of Durmstrang’s language of instruction to be able to follow along in lessons, even if they speak with horrendous accents.
Durmstrang could well have been founded by German-speaking wizards about the same time Hogwarts was founded, but now, thanks to Muggle politics, be dominated to such an extent by Slavic-speakers that the language of instruction itself has changed. Hogwarts might have had similar problems far in the past, since it was apparently founded before 1066; when you’ve got students who speak Gaelic, Anglo-Saxon dialects, Norse and Norman French all in one school, you’ve got to pick out one language and stick with it, however, in those days, the default compromise would certainly have been Latin, as it was in monastic schools.