The study of diffusion in complex networks of one-dimensional (1D) structures embedded within a three-dimensional (3D) domain is essential due to its wide applicability, e.g. modeling blood flow in vascular networks and root-soil interactions. Representing a 3D-3D model with specific interface conditions as a 1D-3D model significantly reduces computational costs while maintaining reliable approximations of the full-dimensional system. However, the analysis and numerical approximation are non-standard due to the singularity of the solution on the 1D lines sources introduced by the dimensional gap. For the numerical approximation of diffusion problems in such coupled 1D-3D systems, discontinuous Galerkin methods (DG) got recent attention, owing to their capabilities for local mesh refinement and high-order local approximation. In this work, we apply local DG methods for a coupled mixed-dimensional diffusion problem in its mixed formulation. By aligning the 1D structures with the 3D mesh and incorporating the coupling between their diffusive quantities directly into the numerical fluxes, the problem is reformulated as a graph representation. Numerical analysis is performed in weighted function spaces and optimal error error estimates are proven and illustrated by numerical simulations for various mixed-dimensional couplings