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Two quantum modes interacting via local couplings to a dissipative bosonic field are investigated theoretically. The model considers two mechanical modes with distinct frequencies coupled optomechanically to the same cavity mode. The dissipative cavity field mediates the interaction between the mechanical modes but also leads to decoherence of the mechanical oscillators. Depending on the ratio between effective interaction strength and dissipation rate, which can be chosen via the pump detuning, the interaction assumes a quantum mechanical or classical character. The distinction between the two regimes is made by the ability of the mediated interaction to correlate the two mechanical modes non-classically. For any cavity decay, there is a regime where the two mechanical modes interact in a non-classical way, which leads us to conclude that optomechanical systems can serve as a model to experimentally study the transition of effective interactions mediated by classical or quantum-mechanical fields. [ABSTRACT FROM AUTHOR]