We consider a Cartesian stochastic variational inequality problem with a monotone map. Monotone stochastic variational inequalities arise naturally, for instance, as the equilibrium conditions of monotone stochastic Nash games over continuous strategy sets or multiuser stochastic optimization problems. We introduce two classes of stochastic approximation methods, each of which requires exactly one projection step at every iteration, and provide convergence analysis for each of them. Of these, the first is a stochastic iterative Tikhonov regularization method which necessitates the update of the regularization parameter after every iteration. The secondmethod is a stochastic iterative proximal-point method, where the centering term is updated after every iteration. The Cartesian structure lends itself to constructing distributed multi-agent extensions and conditions are provided for recovering global convergence in limited coordination variants where agents are allowed to choose their steplength sequences, regularization and centering parameters independently, while meeting a suitable coordination requirement. We apply the proposed class of techniques and their limited coordination versions to a stochastic networked rate allocation problem.