In this paper, we study a privacy-preserving spectrum sharing system to protect secondary users' location privacy while enhancing spectrum access. The location privacy of secondary users can be compromised by an external adversary via the received signal strength (RSS)-based localization technique. To mitigate such privacy threat, we employ a random power perturbation approach that allows each secondary user to judiciously obfuscate the RSS captured by the adversary. While it can protect users' location privacy, the power perturbation approach would inevitably degrade the system performance and bring challenges to the design of the spectrum allocation algorithm. In this work, we adopt a socially-aware database assisted spectrum access system and cast the spectrum allocation under users' power perturbation as a stochastic channel selection game played among the users. To tackle the challenge brought by the privacy protection, we develop a two time-scale distributed learning algorithm, which is shown to converge almost surely to a socially-aware ε-Nash equilibrium. The numerical results show that the higher the privacy protection level is, the more significant the degradation of the network throughput would be.