When sharing sensitive relational databases with other parties, a database owner aims to (i) have privacy guarantees for the database entries, (ii) have liability guarantees (via fingerprinting) in case of unauthorized sharing of its database by the recipients, and (iii) provide a high quality (utility) database to the recipients. We observe that sharing a relational database with privacy and liability guarantees are orthogonal objectives. The former can be achieved by injecting noise into the database to prevent inference of the original data values, whereas, the latter can be achieved by hiding unique marks inside the database to trace malicious parties (data recipients) who redistribute the data without the authorization. We achieve these two objectives simultaneously by proposing a novel entry-level differentially-private fingerprinting mechanism for relational databases. At a high level, the proposed mechanism fulfills the privacy and liability requirements by leveraging the randomization nature that is intrinsic to fingerprinting and achieves desired entry-level privacy guarantees. To be more specific, we devise a bit-level random response scheme to achieve differential privacy guarantee for arbitrary data entries when sharing the entire database, and then, based on this, we develop an $ε$-entry-level differentially-private fingerprinting mechanism. Next, we theoretically analyze the relationships between privacy guarantee, fingerprint robustness, and database utility by deriving closed form expressions. The outcome of this analysis allows us to bound the privacy leakage caused by attribute inference attack and characterize the privacy-utility coupling and privacy-fingerprint robustness coupling. Furthermore, we also propose a SVT-based solution to control the cumulative privacy loss when fingerprinted copies of a database are shared with multiple recipients.