A data curator would like to collect data from privacy aware agents. The collected data will be used for the benefit of all agents. Can the curator incentivize the agents to share their data truthfully? Can he guarantee that truthful sharing will be the unique equilibrium? Can he provide some stability guarantees on such equilibrium? We study necessary and sufficient conditions for these questions to be answered positively and complement these results with corresponding data collection protocols for the curator. Our results account for a broad interpretation of the notion of privacy awareness.