The alternating direction method of multipliers (ADMM) has recently been recognized as a promising approach for large-scale machine learning models. However, very few results study ADMM from the aspect of communication costs, especially jointly with privacy preservation. We investigate the communication efficiency and privacy of ADMM in solving the consensus optimization problem over decentralized networks. We first propose incremental ADMM (I-ADMM), the updating order of which follows a Hamiltonian cycle. To protect privacy for agents against external eavesdroppers, we investigate I-ADMM with privacy preservation, where randomized initialization and step size perturbation are adopted. Using numerical results from simulations, we demonstrate that the proposed I-ADMM with step size perturbation can be both communication efficient and privacy preserving.