We study irreversible processes for nonlinear oscillators networks described by complex-valued Langevin equations that account for coupling to different thermochemical baths. Dissipation is introduced via non-Hermitian terms in the Hamiltonian of the model. We apply the stochastic thermodynamics formalism to compute explicit expressions for the entropy production rates. We discuss in particular the nonequilibrium steady states of the network characterized by a constant production rate of entropy and flows of energy and particle currents. For two specific examples, a one-dimensional chain and a dimer, numerical calculations are presented. The role of asymmetric coupling among the oscillators on the entropy production is illustrated.