Should you copy a person's accent when pronouncing their name?

For example, an English person called Bob might introduce themselves as "Bob", whereas an American person called Bob might introduce themselves as "Bahb". (Sorry, don't know the phonetic alphabet but hopefully you get my gist)

Should you pronounce those two people's names the same, with your own natural accent, or should you copy how the person says their own name?

Edit: I specifically picked a generic English name with different pronunciation across different accents. I know my wording wasn't great, sorry! Hopefully the edit is a bit clearer.