The subject of this paper is the decentralization of decision making when agents have information which is incomplete and possibly exclusive. The first theorem states that in economic environments with three or more individuals, there exists a mechanism whose Bayesian equilibria coincide with a desired collection of social choice functions if and only if closure, incentive compatibility, and Bayesian monotonicity conditions are satisfied. With regards to the previous literature, this closes the gap between necessary and sufficient conditions by using a slightly stronger definition of Bayesian monotonicity, and extends the definition of economic environments to permit externalities. The second theorem extends the analysis to noneconomic environments. It states that closure, incentive compatibility, and a combination of monotonicity and no-veto conditions, are sufficient for implementation, when there are at least three individuals. An example shows that in a Bayesian setting, the second theorem does not hold for separate monotonicity and no-veto style conditions.