In the international development community, the term "digital public goods" is used to describe open-source digital products (e.g., software, datasets) that aim to address the United Nations (UN) Sustainable Development Goals. DPGs are increasingly being used to deliver government services around the world (e.g., ID management, healthcare registration). Because DPGs may handle sensitive data, the UN has established user privacy as a first-order requirement for DPGs. The privacy risks of DPGs are currently managed in part by the DPG standard, which includes a prerequisite questionnaire with questions designed to evaluate a DPG's privacy posture. This study examines the effectiveness of the current DPG standard for ensuring adequate privacy protections. We present a systematic assessment of responses from DPGs regarding their protections of users' privacy. We also present in-depth case studies from three widely-used DPGs to identify privacy threats and compare this to their responses to the DPG standard. Our findings reveal limitations in the current DPG standard's evaluation approach. We conclude by presenting preliminary recommendations and suggestions for strengthening the DPG standard as it relates to privacy. Additionally, we hope this study encourages more usable privacy research on communicating privacy, not only to end users but also third-party adopters of user-facing technologies.