Collaborative spectrum sensing (CSS) has been shown to be able to highly improve the performance of spectrum sensing in cognitive radio networks. However, most existing works focused on either centralized approaches that rely on a global fusion center, thus requiring significant overhead, or on distributed approaches that rely on disjoint coalitions of secondary users (SUs) in which an SU can only cooperate with a single, selected coalition, hence limiting the performance gains of CSS. In this paper, a novel, coalition-based approach to CSS is proposed in which an SU can share its sensing results with more than one coalition. The problem is formulated using a novel class of cooperative games, known as overlapping coalitional games, which enables the SUs to decide, in a distributed manner, on the number of coalitions in which they wish to cooperate, depending on the associated benefit and cost tradeoffs. To solve this game, a novel, distributed algorithm is proposed using which the SUs can self-organize into a stable overlapping coalitional structure. Simulation results show that our proposed algorithm significantly improves the performance in terms of both the average probability of misdetection and the convergence time, relative to the noncooperative case and the state-of-art cooperative CSS with non-overlapping coalitions.