Swelling experiments and large-strain compression tests were performed on Alginate-gelatin (alg-GEL) hydrogels and enhanced derivative alginate di-aldehyde gelatin hydrogels (ADA-GEL). The evolution of the hydrogels’ compositions and their densities were measured. In order to determine average structural parameters of the hydrogel networks, the rubber elasticity theory was applied through direct hyperelastic parameter identification of the Neo-Hookean and Arruda–Boyce models. Both models are based on a macromolecular description of the strain energy density function. Deformation of the swollen polymer networks were also examined through the rubber elasticity theory. Furthermore, the equilibrium-swelling equation was implemented in order to determine the polymer-solvent interaction parameter. The results show that the cross-linking density and form affect the swelling behavior and the stiffness of the hydrogels. Moreover, the stress-strain responses of the hydrogel specimens were marked by hysteresis. The latter quantifies the total dissipated energy during the loading-unloading cycle due to reversible visco-elastic deformation and irreversible thermal dissipation. Based on this, the polymer-water friction coefficient correlates viscous dissipation to the mesh-sizes that was obtained through the rubber elasticity. The mesh-size of the elastic network can thus be regarded as concentration-dependent characteristic length that relates to the viscous nature of the hydrogels.