We propose and analyze a driven-dissipative quantum sensor that is continuously monitored close to a dissipative critical point. The sensor relies on the critical open Rabi model with the spin and phonon degrees of freedom of a single trapped ion to achieve criticality-enhanced sensitivity. Effective continuous monitoring of the sensor with nearly unit efficiency is realized via a co-trapped ancilla ion that switches between dark and bright internal states conditioned on a `jump' of the phonon population. We demonstrate that the critical sensor achieves a scaling beyond the standard shot noise limit under realistic conditions and is robust to experimental imperfections.