This is a general problem whenever there's an alphabet mismatch. Unless there's a 1:1 mapping between phonemes in different languages, one always need to come up with some scheme that will necessarily be imperfect, as seen e.g. when transliterating Slavic or Indian names into English. So long as there is a consistent government-mandated or at least government-blessed system, though, they can work things out fine.

(There's a separate issue here where a system for a specific pair of languages might get codified and become "frozen in time" even as either or both languages evolve. For example, the Russian Polivanov system for transliterating Japanese uses "си" for "シ" because the standard pronunciation of "щ" at the time was more like "шч", similar to Ukrainian, so it was clearly the wrong choice back then - and yet clearly the right choice now if not for backwards compatibility concerns.)