In this paper, we develop a user-centric privacy framework for quantitatively assessing the exposure of personal information in open settings. Our formalization addresses key-challenges posed by such open settings, such as the unstructured dissemination of heterogeneous information and the necessity of user- and context-dependent privacy requirements. We propose a new definition of information sensitivity derived from our formalization of privacy requirements, and, as a sanity check, show that hard non-disclosure guarantees are impossible to achieve in open settings. After that, we provide an instantiation of our framework to address the identity disclosure problem, leading to the novel notion of d-convergence. d-convergence is based on indistinguishability of entities and it bounds the likelihood with which an adversary successfully links two profiles of the same user across online communities. Finally, we provide a large-scale evaluation of our framework on a collection of 15 million comments collected from the Online Social Network Reddit. Our evaluation validates the notion of d-convergence for assessing the linkability of entities in our data set and provides deeper insights into the data set's structure.