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.
Context and other languages
When pronouncing a name from a different language, I firmly believe you should copy the pronunciation of the owner of that name, and not Anglicise the name unless asked to. I say this as a speaker of a language that English people regularly mispronounce and even insist to me that they know the correct pronunciation of my language.
I try to say it exactly like they say it.
I had a coworker named Mahmoud, and all my native-English-speaking coworkers heavily anglicized the pronunciation, removing the throat-clearing sound of the h, and changing the first vowel to like the a in “math” rather than like the a in “mall”. Whenever I spoke to him, I tried to copy his inflection as best I could, and and he seemed delighted, but I never clarified if it was about my pronunciation.