I can't speak for China, but I can speak for Japan. I occasionally transliterate my name for one reason or another, but 99% of the time I'm rocking my name the way I identify with it. Don't mistake me for not participating the local culture. I speak the language, the government (city and federal), and my employer, a local firm, all use my latin name on all official documents and informal communications. My coworkers and friends will usually use my latin spelling as well. Bear in mind, these are usually mixed-provenance sentences where they'll switch to latin for my name in the middle of an otherwise fully native sentence. I'm happy to transliterate it when the situation calls for it but I'd prefer it not be forced on me if it can be helped.

To answer a sibling comment, do I identify it more spoken or written, I would say the written form. I'm tolerant of accents, foreign languages with separate syllabaries, verbal typos, and other such errata, so I don't really care strongly how my name is pronounced as long as an earnest attempt is made.