This paper discusses distributed approaches for the solution of random convex programs (RCPs). RCPs are convex optimization problems with a (usually large) number N of randomly extracted constraints; they arise in several application areas, especially in the context of decision-making under uncertainty; see [G. C. Calafiore, SIAM J. Optim., 20 (2010), pp. 3427-3464; G. C. Calafiore and M. C. Campi, IEEE Trans. Automat. Control, 51 (2006), pp. 742-753]. We here consider a setup in which instances of the random constraints (the scenario) are not held by a single centralized processing unit, but are instead distributed among different nodes of a network. Each node "sees" only a small subset of the constraints, and may communicate with neighbors. The objective is to make all nodes converge to the same solution as the centralized RCP problem. To this end, we develop two distributed algorithms that are variants of the constraints consensus algorithm [G. Notarstefano and F. Bullo, Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, 2007, pp. 927-932; G. Notarstefano and F. Bullo, IEEE Trans. Automat. Control, 56 (2011), pp. 2247-2261]: the active constraints consensus algorithm, and the vertex constraints consensus (VCC) algorithm. We show that the active constraints consensus algorithm computes the overall optimal solution in finite time, and with almost surely bounded communication at each iteration of the algorithm. The VCC algorithm is instead tailored for the special case in which the constraint functions are convex also with respect to the uncertain parameters, and it computes the solution in a number of iterations bounded by the diameter of the communication graph. We further devise a variant of the VCC algorithm, namely quantized vertex constraints consensus (qVCC), to cope with the case in which communication bandwidth among processors is bounded. We discuss several applications of the proposed distributed techniques, including estimation, classification, and random model predictive control, and we present a numerical analysis of the performance of the proposed methods. As a complementary numerical result, we show that the parallel computation of the scenario solution using the active constraints consensus algorithm significantly outperforms its centralized equivalent.