Counting items in a distributed system, and estimating the cardinality of multisets in particular, is important for a large variety of applications and a fundamental building block for emerging Internet-scale information systems. Examples of such applications range from optimizing query access plans in peer-to-peer data sharing, to computing the significance (rank/score) of data items in distributed information retrieval. The general formal problem addressed in this article is computing the network-wide distinct number of items with some property (e.g., distinct files with file name containing “spiderman”) where each node in the network holds an arbitrary subset, possibly overlapping the subsets of other nodes. The key requirements that a viable approach must satisfy are: (1) scalability towards very large network size, (2) efficiency regarding messaging overhead, (3) load balance of storage and access, (4) accuracy of the cardinality estimation, and (5) simplicity and easy integration in applications. This article contributes the DHS (Distributed Hash Sketches) method for this problem setting: a distributed, scalable, efficient, and accurate multiset cardinality estimator. DHSis based on hash sketches for probabilistic counting, but distributes the bits of each counter across network nodes in a judicious manner based on principles of Distributed Hash Tables, paying careful attention to fast access and aggregation as well as update costs. The article discusses various design choices, exhibiting tunable trade-offs between estimation accuracy, hop-count efficiency, and load distribution fairness. We further contribute a full-fledged, publicly available, open-source implementation of all our methods, and a comprehensive experimental evaluation for various settings.