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Despite its many useful theoretical properties, differential privacy (DP) has one substantial blind spot: any release that non-trivially depends on confidential data without additional privacy-preserving randomization fails to satisfy DP. Such a restriction is rarely met in practice, as most data releases under DP are actually "partially private” data (PPD). This poses a significant barrier to accounting for privacy risk and utility under logistical constraints imposed on data curators, especially those working with official statistics. In this paper, we propose a privacy definition which accommodates PPD and prove it maintains similar properties to standard DP. We derive optimal transport-based mechanisms for releasing PPD that satisfy our definition and algorithms for valid statistical inference using PPD, demonstrating their improved performance over post-processing methods. Finally, we apply these methods to a case study on US Census and CDC PPD to investigate private COVID-19 infection rates. In doing so, we show how data curators can use our framework to overcome barriers to operationalizing formal privacy while providing more transparency and accountability to users.