Quantum systems in nonequilibrium conditions, where coherent many-body interactions compete with dissipative effects, can feature rich phase diagrams and emergent critical behavior. Associated collective effects, together with the continuous observation of quanta dissipated into the environment—typically photons—allow one to achieve quantum-enhanced parameter estimation. However, protocols for tapping this enhancement typically involve intricate measurements on the combined system-environment state. Here, we show that many-body quantum enhancement can in fact be obtained through classical measurements such as photon counting and homodyne detection. We illustrate this in detail for a class of open spin-boson models, which can be realized in trapped-ion or cavity QED setups. Our findings highlight a route toward the design of systems that enable a practical implementation of quantum-enhanced metrology through continuous classical measurements.