Just a note: I am definitely not putting down the people who speak their native language with a variation, an accent or a local dialect.
Unlike an accent in a foreign language, which means that one needs to work more on his studies, an "accent" in the *native* language just shows one's heritage.
My native language is Russian, but I have a fairly strong Ukrainian flavor in my speech; I am sure I could get rid of it if I wanted to but I see no reason why I should. Likewise, I am making a good deal of effort to get rid of my Russian accent in English (because it is foreign), but were I from say Alabama, I would not be trying to speak like someone from Chicago.