In this paper, some privacy-preserving features for distributed subgradient optimization algorithms are considered. Most of the existing distributed algorithms focus mainly on the algorithm design and convergence analysis, but not the protection of agents' privacy. Privacy is becoming an increasingly important issue in applications involving sensitive information. In this paper, we first show that the distributed subgradient synchronous homogeneous-stepsize algorithm is not privacy preserving in the sense that the malicious agent can asymptotically discover other agents' subgradients by transmitting untrue estimates to its neighbors. Then a distributed subgradient asynchronous heterogeneous-stepsize projection algorithm is proposed and accordingly its convergence and optimality is established. In contrast to the synchronous homogeneous-stepsize algorithm, in the new algorithm agents make their optimization updates asynchronously with heterogeneous stepsizes. The introduced two mechanisms of projection operation and asynchronous heterogeneous-stepsize optimization can guarantee that agents' privacy can be effectively protected.