In an effort to meet growing demands on the radio frequency spectrum, regulators are exploring methods to enable band sharing among a diverse set of user devices. Proposed spectrum access systems would dynamically assign spectrum resources to users, maintaining databases of spectrum use information. While these systems are anticipated to increase the efficiency of spectrum sharing, incumbent users have raised concerns about exposing details of their operations and have questioned whether their privacy can be protected. In this paper, we explore whether primary users can retain a critical level of privacy when a system uses their information to enable dynamic access to the spectrum by other users. Under a variety of operational scenarios and user models, we examine adversary techniques to exploit the spectrum access system and obfuscation strategies to protect user privacy. We also develop analytical methods to quantify the performance of both the adversary and obfuscation strategies. To our knowledge, this is the first work that considers the privacy of a primary user in the setting of a highly dynamic spectrum access system. Privacy analysis of this kind will help to enable adoption of shared spectrum access systems by allowing incumbent users to quantify and mitigate risks to their privacy.