Yes, the United States recognizes dual nationality. I was talking to the guy next to me on a plane one time and he said that dual nationality was not recognized in the U.S. or China. After coming home I forgot about the conversation, but I remembered it when I thought about a friend who is a U.S. and Chinese citizen. In the U.S. she's a citizen of this country and of China, but according to China she is only a citizen of the U.S.
I wonder how Chinese who are citizens of other countries become citizens of China again when they return. This is a situation my friend's family will have to deal with when they come become citizens of the U.S., but they made it sound like an easy process. Maybe it is as simple as denying citizenship of the other country and claiming citizenship of China.