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 would probably be a bit offended and think you were making fun of my accent if you copied me.
I’d be happy they made the effort to try.
A name is just the noise another person makes to get your attention or address you. If they make the wrong noise - it’s not gonna work as well.
They can try . . . by saying my name.
You don’t need to copy my Australian accent to say my name.