My take is that all people in Starfleet speak a common language (be it a future version of English, Vulcan, Galaxyan, or whatever), which is conveyed as English in the show. Universal translators can fail, and no sensible organization would make itself susceptible to such a critical failure if possible. All Federation officials would probably speak this language too, to ease communications in government, and the same applies to most Federation citizens. Being multilingual is probably quite common. Local languages (what we would call national languages) can be still used at home with family and friends, but not in planetary or interplanetary communications. It won't be much different from current countries with more than one official languages.