Controlling large-scale cyber-physical systems necessitates optimal distributed policies, relying solely on local real-time data and limited communication with neighboring agents. However, finding optimal controllers remains challenging, even in seemingly simple scenarios. Parameterizing these policies using Neural Networks (NNs) can deliver good performance, but their sensitivity to small input changes can destabilize the closed-loop system. This paper addresses this issue for a network of nonlinear dissipative systems. Specifically, we leverage well-established port-Hamiltonian structures to characterize deep distributed control policies with closed-loop stability guarantees and a finite L2 gain, regardless of specific NN parameters. This eliminates the need to constrain the parameters during optimization and enables training with standard methods like stochastic gradient descent. A numerical study on the consensus control of Kuramoto oscillators demonstrates the effectiveness of the proposed controllers.