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.
This is my approach. You are gonna get a kind of Americanized version because I can’t help it, but:
We have an Irinya in our department, soft r, I sounds like our long E - I am not gonna call her Irine! Soft R I am familiar with from Spanish, I’m sure it’s not spot on as she is Eastern European but close enough for her, she says it sounds right. One lady on our team can’t even hear what the R sounds like and can’t make the sound, she calls her eeReenya with the hard R. Still acceptable to Irinya, that’s what matters.
If it’s some sounds I can’t pronounce, I’d ask what they want to be called if I can’t make those sounds but in general yes I try to pronounce people’s names as they do, to the extent I can.