We consider a network of autonomous agents whose outputs are actions in a game with coupled constraints. In such network scenarios, agents seeking to minimize coupled cost functions using distributed information while satisfying the coupled constraints. Current methods consider the small class of multi-integrator agents using primal-dual methods. These methods can only ensure constraint satisfaction in steady-state. In contrast, we propose an inexact penalty method using a barrier function for nonlinear agents with equilibrium-independent passive dynamics. We show that these dynamics converge to an epsilon-GNE while satisfying the constraints for all time, not only in steady-state. We develop these dynamics in both the full-information and partial-information settings. In the partial-information setting, dynamic estimates of the others' actions are used to make decisions and are updated through local communication. Applications to optical networks and velocity synchronization of flexible robots are provided.