This study concerns the risks of privacy disclosure when sharing and releasing a dataset in which each individual may be associated with multiple records. Existing data privacy approaches and policies typically assume that each individual in a shared dataset corresponds to a single record, leading to an underestimation of the disclosure risks in multiple records per person scenarios. We propose two novel measures of privacy disclosure to arrive at a more appropriate assessment of disclosure risks. The first measure assesses individual-record disclosure risk based upon the frequency distribution of individuals' occurrences. The second measure assesses sensitive-attribute disclosure risk based upon the number of individuals affiliated with a sensitive value. We show that the two proposed disclosure measures generalize the well-known -anonymity and -diversity measures, respectively, and work for scenarios with either a single record or multiple records per person. We have developed an efficient computational procedure that integrates the two proposed measures and a data quality measure to anonymize the data with multiple records per person when sharing and releasing the data for research and analytics. The results of the experimental evaluation using real-world data demonstrate the advantage of the proposed approach over existing techniques for protecting privacy while preserving data quality.