Recently, collaborative distributed energy management systems (CoDEMS) have emerged as an effective solution to manage distributed energy resources in microgrid. In CoDEMS, devices collaborate in a distributive manner over communication networks to meet electrical loads and supply balance at minimum cost. However, mutual information exchanges among the devices in CoDEMS may leak important information about the devices states. In this paper, we investigate the challenging problem of how to achieve optimality while preserving the privacy of CoDEMS at relatively low cost. Unlike many previous works that preserve the privacy by using additive noises, we propose a novel random-weight privacy-preserving algorithm with error compensation, termed as REP-CoDEMS, for CoDEMS. In the proposal, each distributed device generates two random weights each time and it communicates with its neighbor conveying values based on the weights, incremental cost estimation and power imbalance estimation information along with a novel error compensation term to eliminate the error induced by the random weights. We theoretically prove that the proposed REP-CoDEMS algorithm converges and preserves the privacy of all devices. We also derive analytical expressions of the maximum privacy disclosure probability for initial and final states of the CoDEMS. In addition, we conduct extensive simulations and the results demonstrate the effectiveness of the proposed algorithm.