Regarding Austria-Hungary:

Each regiment had its own regimental language(s) - if at least 20% of the rank-and-file inside the regiment spoke a certain language, then it had to be learned by the commissioned officers as well. Officers had to prove their language capabilities in front of a special commission three years after the officer entered his unit and could not achieve further promotion without mastering the languages of their regiment, so there was a rather strong incentive for the officers to learn the languages as soon as possible.

Also it's worth noting that military academies placed emphasis on learning languages, for example in the Theresian Military Academy in Wiener Neustadt in late 19th century one learned (aside from German) French, Czech and one of either Polish or Hungarian; though the quality of the language lectures wasn't ideal. The army itself did not help officers with learning languages all that much, so an individual officer had to learn it more or less by himself, though he often got help from NCOs who were familiar with the regimental language.

Regarding the communication of officers with rank-and-file then aside from officers knowing the regimental languages, everyone had to master a set of 80 basic commands in German (stuff like "forwards!", "halt!" or "fire!"). However there were also cases of a regiment developing a special military jargon or even artificial rump language that consisted of mashing several languages together; the most prominent was the so-called "Armee-Slawisch" ("Army Slavic") language which was a mix of predominantly Czech (though there were words from other Slavic languages as well) and German words, that was often used as a "lingua franca" among Slavic-speaking troops.

All in all, it could be said that A-H managed the language barriers relatively well, though it started having serious problems in WWI, when due to the high casualty rate among the officer corps, it had to be filled with reserve officers with inadequate language skills and there are individual accounts of for example soldiers staying together in a foxhole in a complete silence as they could not understand each other.

Source:

DEÁK, István: Beyond Nationalism: A Social and Political History of the Habsburg Officer Corps, 1848-1918; 1990