Regenerative medicine aims to restore damaged tissues and mainly takes advantage of human mesenchymal stromal cells (hMSCs), either alone or combined with three-dimensional scaffolds. The scaffold is generally considered a support, and its contribution to hMSC proliferation and differentiation is unknown or poorly investigated. The aim of this study was to evaluate the capability of an innovative three-dimensional gelatin–chitosan hybrid hydrogel scaffold (HC) to activate the osteogenic differentiation process in hMSCs. We seeded hMSCs from adipose tissue (AT-hMSCs) and bone marrow (BM-hMSCs) in highly performing HC of varying chitosan content in the presence of growing medium (GM) or osteogenic medium (OM) combined with Fetal Bovine Serum (FBS) or human platelet lysate (hPL). We primarily evaluated the viability and the proliferation of AT-hMSCs and BM-hMSCs under different conditions. Then, in order to analyse the activation of osteogenic differentiation, the osteopontin (OPN) transcript was absolutely quantified at day 21 by digital PCR. OPN was expressed under all conditions, in both BM-hMSCs and AT-hMSCs. Cells seeded in HC cultured with OM+hPL presented the highest OPN transcript levels, as expected. Interestingly, both BM-hMSCs and AT-hMSCs cultured with GM+FBS expressed OPN. In particular, BM-hMSCs cultured with GM+FBS expressed more OPN than those cultured with GM+hPL and OM+FBS; AT-hMSCs cultured with GM+FBS presented a lower expression of OPN when compared with those cultured with GM+hPL, but no significant difference was detected when compared with AT-hMSCs cultured with OM+FBS. No OPN expression was detected in negative controls. These results show the capability of HC to primarily and independently activate osteogenic differentiation pathways in hMCSs. Therefore, these scaffolds may be considered no more as a simple support, rather than active players in the differentiative and regenerative process.