We consider distributed convex optimization problems that involve a separable objective function and nontrivial functional constraints, such as linear matrix inequalities. We propose a decentralized and computationally inexpensive algorithm, which is based on the concept of approximate projections. Our algorithm is one of the consensus-based methods in that, at every iteration, each agent performs a consensus update of its decision variables followed by an optimization step of its local objective function and local constraints. Unlike other methods, the last step of our method is not a Euclidean projection onto the feasible set, but instead a subgradient step in the direction that minimizes the local constraint violation. We propose two different averaging schemes to mitigate the disagreements among the agents' local estimates. We show that the algorithms converge almost surely, i.e., every agent agrees on the same optimal solution, under the assumption that the objective functions and constraint functions are nondifferentiable and their subgradients are bounded. We provide simulation results on a decentralized optimal gossip averaging problem, which involves semidefinite programming constraints, to complement our theoretical results.