Many-body effects are at the very heart of diverse phenomena found in condensed-matter physics. One striking example is the Mott-insulator phase, where conductivity is suppressed as a result of a strong repulsive interaction. Advances in cold-atom physics have led to the realization of the Mott-insulating phases of atoms in an optical lattice, mimicking the corresponding condensed-matter systems. Here, we explore an exotic strongly-correlated system of interacting dual Mott insulators of bosons and fermions. We find that an interspecies interaction between bosons and fermions drastically modifies each of the Mott insulators, causing effects that include melting, generation of composite particles, an anti-correlated phase and complete phase separation. Comparison between the experimental results and numerical simulations indicate intrinsic adiabatic heating and cooling for the attractively and repulsively interacting dual Mott insulators, respectively.