We address the question of verifying the quantumness of thermal machines. A Szilárd engine is truly quantum if its work output cannot be described by a local hidden state model, i.e., an objective local statistical ensemble. Quantumness in this scenario is revealed by a steering-type inequality which bounds the classically extractable work. A quantum Maxwell demon can violate that inequality by exploiting quantum correlations between the work medium and the thermal environment. While for a classical Szilárd engine an objective description of the medium always exists, any such description can be ruled out by a steering task in a truly quantum case.