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.
It’s normal to come across words you don’t know how to pronounce in your native language. When you do, you either ask someone for the pronunciation or stumble along until someone corrects you. Names are no different and you shouldn’t be made to feel ashamed by any earnest attempt to pronounce them. Forget about the accent and focus on the specific sounds involved in the name. If you cannot form sounds that are foreign to you, the owner of the name will likely help you pronounce it in the language you’re comfortable with. They are probably used to doing this.