We study noninteracting fermionic systems undergoing continuous monitoring and driven by biased reservoirs. Averaging over the measurement outcomes, we derive exact formulas for the particle and heat flows in the system. We show that these currents feature competing elastic and inelastic components, which depend nontrivially on the monitoring strength γ. We highlight that monitor-induced inelastic processes lead to nonreciprocal currents, allowing one to extract work from measurements without active feedback control. We illustrate our formalism with two distinct monitoring schemes providing measurement-induced power or cooling. Optimal performances are found for values of the monitoring strength γ, which are hard to address with perturbative approaches.