We consider thermal machines powered by locally equilibrium reservoirs that share classical or quantum correlations. The reservoirs are modeled by the so-called collisional model or repeated interactions model. In our framework, two reservoir particles, initially prepared in a thermal state, are correlated through a unitary transformation and afterward interact locally with the two quantum subsystems which form the working fluid. For a particular class of unitaries, we show how the transformation applied to the reservoir particles affects the amount of heat transferred and the work produced. We then compute the distribution of heat and work when the unitary is chosen randomly, proving that the total swap transformation is the optimal one. Finally, we analyze the performance of the machines in terms of classical and quantum correlations established among the microscopic constituents of the machine.