Mussels secrete protein-based byssal threads to tether to rocks, ships, and other organisms underwater. The secreted marine mussel adhesive proteins (MAPs) contain the peculiar amino acid L-3,4-dihydroxyphenylalanine (DOPA), whose catechol group content contributes greatly to their outstanding adhesive properties. Inspired by such mussel bioadhesion, we demonstrate that catechol-modified polysaccharides can be used to obtain adhesive membranes using the compaction of polyelectrolyte complexes (CoPEC) method. It is a simple and versatile approach that uses polyelectrolyte complexes as building blocks that coalesce and dry as membrane constructs simply as a result of sedimentation and mild temperature. We used two natural and biocompatible polymers: chitosan (CHI) as a polycation and hyaluronic acid (HA) as a polyanion. The CoPEC technique also allowed the entrapment of ternary bioactive glass nanoparticles to stimulate mineralization. Moreover, combinations of these polymers modified with catechol groups were made to enhance the adhesive properties of the assembled membranes. Extensive physico-chemical characterization was performed to investigate the successful production of composite CoPEC membranes in terms of surface morphology, wettability, stability, mechanical performance, in vitro bioactivity, and cellular behavior. Considering the promising properties exhibited by the obtained membranes, new adhesives suitable for the regeneration of hard tissues can be envisaged.