There are plenty of US tech companies that have success in China.
For example I often use Agoda or hotels.com when I travel in China.
I think AirBnB will have plenty of competition in China and broader Asia from well-established players such as CTrip, Agoda or Rakuten.
The "social network" (rating your visitors etc.) aspect of AirBnB I think will matter less in those parts of the world ... but maybe I am wrong and things will change.