No trade theorems have shown that new information will not lead to trade when agents share the same prior beliefs. This paper explores the structure of no trade theorems with heterogeneous prior beliefs. It is shown how different notions of efficiency under asymmetric information--ex ante, interim, ex post--are related to agents' prior beliefs, as well as incentive compatible and public versions of those efficiency concepts. These efficiency results are used to characterize necessary and sufficient conditions on agents' beliefs for no trade theorems in different trading environments.