DS9's Little Green Men certainly seems to imply that at the very least Quark, Rom, and Nog are speaking the Ferengi language and rely on their UTs to understand everyone else and to be understood. The Japanese guy in Voyager's The 37s was speaking English though he claimed to hear everyone else speaking Japanese. Presumably everyone is speaking their native languages.
I'm sure there are exceptions. Given Spock is half human and chose to work among humans he probably has learned English. Worf should actually be speaking Russian given he was raised by Russian parents. Though since they are always speaking English it stands to reason he learned the language. T'Pol likely was actually speaking English given UTs weren't as common back in the day.
Mr. B wrote:
In the episode "Code of Honor," Data makes a comment that could lead the viewer to believe the French language is as dead as Latin. Picard's flawless English, English accent and affinity for English literature point to that as well. French is a dying language today.
Actually, all Data said was that French was "archaic." Which implies it isn't a dead tongue, just not a common one.