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.
To add to this, accept when people tell you it’s fine to pronounce their name wrong. My name is very hard to pronounce if you don’t speak my native language, and I prefer that people mispronounce it the “obvious” way, instead of trying to approximate it because then I have people calling me by 20 different variations, and sometimes I’ve no idea they are referring to me.