We consider a demand management problem of an energy community, in which several users obtain energy from an external organization such as an energy company, and pay for the energy according to pre-specified prices that consist of a time-dependent price per unit of energy, as well as a separate price for peak demand. Since users' utilities are private information which they may not be willing to share, a mediator, known as the planner, is introduced to help optimize the overall satisfaction of the community (total utility minus total payments) by mechanism design. A mechanism consists of message spaces, and a set of tax and allocation functions for each user. Once we implement the mechanism, each user reports a message chosen from her own message space, and then receives some amount of energy determined by the allocation function and pays the tax specified by the tax function. A desirable mechanism induces a game in which Nash equilibria (NE) provides an allocation that coincides with the optimal allocation for the community. As a starting point, we design a standard, "centralized" mechanism for the energy community with desirable properties such as full implementation, strong budget balance and individual rationality for both users and the planner. Then we extend this mechanism to the case of communities where message exchanges only happen among neighborhoods, and consequently, the tax and allocation functions of each user are only determined by the messages from her neighbors. All the properties designed for the centralized mechanism are preserved in the distributed mechanism. Subsequently, we address the problem of learning guarantees for both mechanisms. One set of learning dynamic was found in this primary work, that guarantees users will reach NE if they follow these dynamics.