Soft biopolymer networks play pivotal roles in governing cellular mechanics, tissue structure, and physiological processes such as blood coagulation. Understanding their permeability and mechanical responses under compression is crucial for elucidating mass transport phenomena and their impact on extra- and intra-cellular behavior as well as processes affecting functionality of blood clots, cartilage and other fibrous tissues. The nonlinear responses of these networks to mechanical stresses prevent application of established linear poro-elasticity models. Despite extensive studies of fibrous network viscoelastic properties under shear deformations, their dynamic responses to compressive deformations remain poorly understood, particularly in physiological contexts of growth and collective migration of solid bodies. Conventional experimental techniques face challenges in accurately evaluating the permeability of these networks, hindering comprehensive understanding of their poromechanical behavior. In this study, we employ a novel poroelastic hybrid approach combining rheometer-based compression rheology with camera-facilitated sample shape detection to directly measure fluid flux and network permeability under controlled compressive strains. Accompanying experimental investigations, a continuum model implemented in finite elements, and an analytical model are developed to interpret the findings. The experimental data align well with the analytical model, revealing the emergence and disappearance of distinct densification regimes within the gel under mechanical stress. This study advances our understanding of the intricate interplay between mechanical forces, fluid flow, and structural properties in soft biopolymer networks, with a specific focus on fibrin- and collagen-based gels which represent the most abundant protein networks in the extracellular environment.