Future applications in environmental monitoring, delivery of services, and transportation of goods motivate the study of deployment and partitioning tasks for groups of autonomous mobile agents. These tasks may be achieved by recent coverage algorithms, based upon the classic methods by Lloyd. These algorithms, however, rely upon critical requirements on the communication network: information is exchanged synchronously among all agents and long-range communication is sometimes required. This work proposes novel coverage algorithms that require only gossip communication, i.e., asynchronous, pairwise, and possibly unreliable communication. Which robot pair communicates at any given time may be selected deterministically or randomly. A key innovative idea is describing coverage algorithms for robot deployment and environment partitioning as dynamical systems on a space of partitions. In other words, we study the evolution of the regions assigned to each agent rather than the evolution of the agents' positions. The proposed gossip algorithms are shown to converge to centroidal Voronoi partitions under mild technical conditions. Our treatment features a broad variety of results in topology, analysis, and geometry. First, we establish the compactness of a suitable space of partitions with respect to the symmetric difference metric. Second, with respect to this metric, we establish the continuity of various geometric maps, including the Voronoi diagram as a function of its generators, the location of a centroid as a function of a set, and the widely known multicenter function studied in facility location problems. Third, we prove two convergence theorems for dynamical systems on metric spaces described by deterministic and stochastic switches.