We use a generalized version of the Myerson-Satterthwaite theorem to study inefficiencies in bilateral bargaining over trade of an indivisible good, where there is two sided private information on the valuations. We show that when preferences are convex and quasi linear, and when the private information represents the magnitude of the utility gain or loss and follows a uniform distribution, that the most efficient mechanism always exhibits a bias towards the status quo. In the case that utility functions are quadratic in the amount traded, we prove that for any incentive compatible direct mechanism, there is an expected bias towards the disagreement point. In other words, for the class of preferences we study, there is a strategic advantage to property rights in the Coase bargaining setup in the presence of incomplete information.