This isn't the case with everyone, but you'll see in a few cases, such as with Captain Georgiou (or Emperess Georgiou) in Star Trek: Discovery, that accents are clearly audible. This is a very closely related question, in which I asked why Chekov spoke English in ST: 2009 with such a heavy accent, instead of just speaking Russian.
If people are uncomfortable with other languages, why can't they speak in their native languages and let the Universal Translator translate for them? And if they are speaking their native tongue, why does the UT create an accent?