Multi-physics mathematical models of the vascular microenvironment assess the impact of microcirculation through the tissue. Since the embedded vascular network is characterized by high complexity, numerical solutions to the related multiscale problem present significant computational challenges. To address them efficiently, we present an alternative strategy that is applied to a mixed-dimensional microcirculation model, based on parametrized partial differential equations (PDEs). The full-order model (FOM) describes microvascular blood flow and is decoupled into a 3D model for the tissue and a 1D model in the vascular network. We propose a multiscale strategy that integrates nonoverlapping domain decomposition (DD) methods [1] with reduced order models (ROM) [2] to handle complex microvascular network simulations. The 3D domain is decomposed into small voxel components, where local numerical solutions are retrieved exploiting non-intrusive and nonlinear ROMs to approximate the parameter-to-solution map in each voxel. We leverage proper orthogonal decomposition (POD) methods and sparse Mesh-Informed Neural Networks (MINNs) to handle the spatial dependencies in the solutions and in the geometrical input data encoding the small-scale features of the embedded microstructure. More precisely, the adopted ROM is built in a supervised learning framework, relying on local finite-element approximations. Firstly, it approximates the POD reduced basis coefficient using a neural network that combines dense layers for the physical parametrization with a MINN for the geometrical input. Then, a closure model based on the geometrical input is added to augment the first approximation, acting as a fine-scale corrector in each subdomain. The global solution is retrieved by assembling the local ROM approximations through a parallel Robin-Robin domain decomposition approach, before updating the 1D solution and iterating the method until convergence. We exploit this methodology to improve traditional homogenization methods for multiscale modeling that capture only the global behavior in each voxel. [1] Lions, P.L. On the Schwarz alternating method III: a variant for nonoverlapping subdomains. Third International Symposium on Domain Decomposition Methods for Partial Differential Equations. SIAM, (1990). [2] Vitullo, P. et al. Nonlinear model order reduction for problems with microstructure using mesh informed neural networks. Finite Elements in Analysis and Design (2024).