To quantify trade-offs between increasing demand for open data sharing and concerns about sensitive information disclosure, statistical data privacy (SDP) methodology analyzes data release mechanisms which sanitize outputs based on confidential data. Two dominant frameworks exist: statistical disclosure control (SDC), and more recent, differential privacy (DP). Despite framing differences, both SDC and DP share the same statistical problems at its core. For inference problems, we may either design optimal release mechanisms and associated estimators that satisfy bounds on disclosure risk, or we may adjust existing sanitized output to create new optimal estimators. Both problems rely on uncertainty quantification in evaluating risk and utility. In this review, we discuss the statistical foundations common to both SDC and DP, highlight major developments in SDP, and present exciting open research problems in private inference.