With the advance in mobile computing, Internet of Things, and ubiquitous wireless connectivity, social sensing-based edge computing (SSEC) has emerged as a new computation paradigm where people and their personally owned devices collect sensor measurements from the physical world and process them at the edge of the network. This article focuses on a privacy-aware task allocation problem where the goal is to optimize the computation task allocation in SSEC systems while respecting the users' customized privacy settings. It introduces a novel game-theoretic privacy-aware task allocation (G-PATA) framework to achieve the goal. G-PATA includes: 1) a bottom-up game-theoretic model to generate the maximum payoffs at end devices while satisfying the end user's privacy settings and 2) a top-down incentive scheme to adjust the rewards for the tasks to ensure that the task allocation decisions made by end devices meet the Quality-of-Service (QoS) requirements of the applications. Furthermore, the framework incorporates an efficient load balancing and iteration reduction component to adapt to the dynamic changes in status and privacy configurations of end devices. The G-PATA framework was implemented on a real-world edge computing platform that consists of heterogeneous end devices (Jetson TX1 and TK1 boards, and Raspberry Pi3). We compare G-PATA with state-of-the-art task allocation schemes through two real-world social sensing applications. The results show that G-PATA significantly outperforms existing approaches under various privacy settings (our scheme achieved as much as 47% improvements in delay reduction for the application and 15% more payoffs for end devices compared to the baselines).