This paper develops algorithms to train linear support vector machines (SVMs) when training data are distributed across different nodes and their communication to a centralized node is prohibited due to, for example, communication overhead or privacy reasons. To accomplish this goal, the centralized linear SVM problem is cast as the solution of coupled decentralized convex optimization subproblems with consensus constraints on the parameters defining the classifier. Using the method of multipliers, distributed training algorithms are derived that do not exchange elements from the training set among nodes. The communications overhead of the novel approach is fixed and fully determined by the topology of the network instead of being determined by the size of the training sets as it is the case for existing incremental approaches. An online algorithm where data arrive sequentially to the nodes is also developed. Simulated tests illustrate the performance of the algorithms.