We consider the problem of determining the top-$k$ largest measurements from a dataset distributed among a network of $n$ agents with noisy communication links. We show that this scenario can be cast as a distributed convex optimization problem called sample quantile inference, which we solve using a two-time-scale stochastic approximation algorithm. Herein, we prove the algorithm's convergence in the almost sure sense to an optimal solution. Moreover, our algorithm handles noise and empirically converges to the correct answer within a small number of iterations.