This paper presents the development of algorithms for mass-energy constrained neural network (MECNN) models that can exactly conserve the overall mass and energy of distributed chemical process systems, even though the noisy steady-state/transient data used for optimal model training violate the same. For developing dynamic mass-energy constrained network models for distributed systems, hybrid series and parallel dynamic-static neural networks are used as candidate architectures. The proposed approaches for solving both the inverse and forward problems are validated considering both steady-state and dynamic data in presence of various noise characteristics. The proposed network structures and algorithms are applied to the development of data-driven models of a nonlinear non-isothermal reactor that involves an exothermic reaction making it significantly challenging to exactly satisfy the mass and energy conservation laws of the system only by using the available input and output boundary conditions.