Some names, like "Michael" sound dramatically different depending on whether you pronounce them the German or English way. Which one do you use when talking in English?
I suspect this is less an English question than a cultural and/or regional question.
Americans tend to pronounce people's names in the way they are introduced, increasingly so over the past few decades as more people have names which derive from other languages, popular culture, or the parents' imagination. However there are regions where this is practiced, and other regions where it's not as common. We like to make fun of the American South (actually more the Southeast) as more "backwards" and xenophobic, but really you might find people unable to pronounce a name correctly in any part of the country.
As anywhere, younger people might be more open to pronouncing unfamiliar names (as best they can) in the same way the owner says it. Older people might be less open-minded about it, or less able to make the necessary sounds.
For someone with a particularly difficult name, especially one with an unusual spelling, if they have lived in the US for a while they will likely adopt a nickname to make their lives easier. For example, I know a guy whose name is written "Xavier", but with the X pronounced not like "Z" (as in xylophone) or "Ex" (as in X-ray) or even as like a rough "J" sound (like the Spanish name "Javier"), but instead as "Ch". So he just introduces himself as "Chavi".
In general, the news media tries to use the foreign pronunciation (not always successfully) for foreign people. I think that to be good practice. If that person is assimilating into US or British society, and they themselves use the English pronunciation of their name, then it would be proper to follow suit.