When I studied Chinese, our teachers gave everyone a Chinese name. Some of the students were Chinese-Americans and already had such a name, although perhaps in a regional dialect such as Cantonese, so the teachers converted those names to Mandarin. My Chinese surname is Sang, written with the character for "mulberry."

I don't know any Westerner who uses a Chinese name when speaking a European language, but it just seems natural when speaking Mandarin.

One of the first things we learned when I studied Japanese was how to write our names in katakana. There are standard substitutions for sounds that do not exist in Japanese, so that "v" is transcribed as "b." The only final consonant allowed in a syllable is "n," so my family name is Sa-n-do-ne-su in katakana, but I use the katakana form of my name only when speaking Japanese.