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.
There’s way too many languages and dialects with way too many sounds out there for this to be practically doable. For foreign names some basic degree of approximation is desirable, but nothing more than that. In principle you shouldn’t expect or demand people to produce sounds not found in their native dialect (unless they’re actually learning the foreign language, but even then they will usually stick to the same language within the same sentence).
Besides, it’s not even odd for people not to be able to pronounce stuff according to the standard norm of their own native language, due to the dialectal variety within the same language.
As for names from within the same language, it could sound artificial and even condescending if you tried to go for a pronunciation not native to you. Bob is just Bob, no need to stress that he’s “American/British Bob”.