In order to receive personalized services, individuals share their personal data with a wide range of service providers, hoping that their data will remain confidential. Thus, in case of an unauthorized distribution of their personal data by these service providers (or in case of a data breach) data owners want to identify the source of such data leakage. Digital fingerprinting schemes have been developed to embed a hidden and unique fingerprint into shared digital content, especially multimedia, to provide such liability guarantees. However, existing techniques utilize the high redundancy in the content, which is typically not included in personal data. In this work, we propose a probabilistic fingerprinting scheme that efficiently generates the fingerprint by considering a fingerprinting probability (to keep the data utility high) and publicly known inherent correlations between data points. To improve the robustness of the proposed scheme against colluding malicious service providers, we also utilize the Boneh-Shaw fingerprinting codes as a part of the proposed scheme. Furthermore, observing similarities between privacy-preserving data sharing techniques (that add controlled noise to the shared data) and the proposed fingerprinting scheme, we make a first attempt to develop a data sharing scheme that provides both privacy and fingerprint robustness at the same time. We experimentally show that fingerprint robustness and privacy have conflicting objectives and we propose a hybrid approach to control such a trade-off with a design parameter. Using the proposed hybrid approach, we show that individuals can improve their level of privacy by slightly compromising from the fingerprint robustness. We implement and evaluate the performance of the proposed scheme on real genomic data. Our experimental results show the efficiency and robustness of the proposed scheme.