Most people do not work on their accents. Some achieve very good, even perfect, accents after living many years in their new country or by using their new language for many years.
However, this is not automatic. There are many people who spend years using a language and never get rid of a heavy accent.
Often, it doesn’t matter to the person. If they live in an immigrant community, sometimes it is even a negative thing to achieve a good accent in the new language. This is because friends and family might think that one is trying to forget their origins.