We consider a distributed constrained convex optimization problem over a multi-agent (no central coordinator) network. We propose a completely decentralized and asynchronous gossip-based random projection (GRP) algorithm that solves the distributed problem using only local communications and computations. We analyze the convergence properties of the algorithm for a diminishing and a constant stepsize which are uncoordinated among agents. For a diminishing stepsize, we prove that the iterates of all agents converge to the same optimal point with probability 1. For a constant stepsize, we establish an error bound on the expected distance from the iterates of the algorithm to the optimal point. We also provide simulation results on a distributed robust model predictive control problem.