We present the study of the dynamics of a two-ring waveguidestructure with space-dependent coupling, constant linear gain and nonlinear absorption. This system can be implemented in various physical situations such as optical waveguides, atomic Bose-Einstein condensates, polarization condensates, etc. It is described by two coupled nonlinear Schrödinger equations. For numerical simulations we take local Gaussian coupling (single-Gaussian and double-Gaussian). We find that, depending on the values of involved parameters, we can obtain several interesting nonlinear phenomena, which include spontaneous symmetry breaking, modulational instability leading to generation of stable circular flows with various vorticities, stable inhomogeneous states with interesting structure of currents flowing between rings, as well as dynamical regimes having signatures of chaotic behavior. In this paper, we only focused on consider phenomenon of spontaneous symmetry breaking in the case of space dependent coupling. The results show that in the case of a coupling between the two rings is a function of single-Gaussian symmetry breaking only between rings. In contrast, in the case of a coupling between them as a double-Gaussian function, the symmetry breaking occurs only in each ring, breaking the symmetry of the space dependent coupling.