Machine Learning (ML) has seen a great potential to solve many power system problems along with its transition into Smart Grid. Specifically, electric distribution systems have witnessed a rapid integration of distributed energy resources (DERs), including photovoltaic (PV) panels, electric vehicles (EV), and smart appliances, etc. Electricity consumers, equipped with such DERs and advanced metering/sensing/computing devices, are becoming self-interested prosumers who can behave more actively for their electric energy consumption. In this paper, the potential of distributed ML in solving the energy trading problem among prosumers of a future electric distribution system - building DC grid cell, is explored, while considering the limited computation, communication, and data privacy issues of the edge entities. A fully distributed energy trading framework based on ML is proposed to optimize the load and price prediction accuracy and energy trading efficiency. Computation resource allocation, communication schemes, ML task scheduling, as well as user sensitive data preserving issues in the distributed ML framework are addressed with consideration of all the economic and physical constraints of the electric distribution systems.