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 saw something about this the other day, but I forget exactly where. They spoke about two famous people, both with given name “Craig” where one was British and the other American. They said that they would deliberately pronounce the name differently for each person in order to reflect that person’s preferred pronunciation.
Approximating that within your own accent wasn’t mentioned, but I assume that would be acceptable.
Another one that springs to mind is the name “Colin”. There was that well-known US politician who insisted that his name was to be pronounced with a long ‘o’ not a short one, which deviated even from the standard US pronunciation.
If I remember correctly, he insisted that if it was to be pronounced the other way, it should have had two L’s in it. Makes me wonder how he spelled/pronounced travel(l)ing.