Scaffolds for tissue engineering are expected to respond to a challenging combination of physical and mechanical requirements, guiding the research towards the development of novel hybrid materials. This study introduces innovative three-dimensional bioresorbable scaffolds, in which a stiff poly(lactic acid) lattice structure is meant to ensure temporary mechanical support, while a bioactive gelatin–chitosan hydrogel is incorporated to provide a better environment for cell adhesion and proliferation. The scaffolds present a core–shell structure, in which the lattice core is realized by additive manufacturing, while the shell is nested throughout the core by grafting and crosslinking a hydrogel forming solution. After subsequent freeze-drying, the hydrogel network forms a highly interconnected porous structure that completely envelops the poly(lactic acid) core. Thanks to this strategy, it is easy to tailor the scaffold properties for a specific target application by properly designing the lattice geometry and the core/shell ratio, which are found to significantly affect the scaffold mechanical performance and its bioresorption. Scaffolds with a higher core/shell ratio exhibit higher mechanical properties, whereas reducing the core/shell ratio results in higher values of bioactive hydrogel content. Hydrogel contents up to 25 wt% could be achieved while maintaining high compression stiffness (>200 MPa) and strength (>5 MPa), overall, within the range of values displayed by human bone tissue. In addition, mechanical properties remain stable after prolonged immersion in water at body temperature for several weeks. On the other hand, the hydrogel undergoes gradual and homogeneous degradation over time, but the core–shell integrity and structural stability are nevertheless maintained during at least 7-week hydrolytic degradation tests. In vitro experiments with human mesenchymal stromal cells reveal that the core–shell scaffolds are biocompatible, and their physical–mechanical properties and architecture are suitable to support cell growth and osteogenic differentiation, as demonstrated by hydroxyapatite formation. These results suggest that the bioresorbable core–shell scaffolds can be considered and further studied, in view of clinically relevant endpoints in bone regenerative medicine.