The interplay between information-scrambling Hamiltonians and local continuous measurements hosts platforms for exotic measurement-induced phase transition in out-of-equilibrium steady states. Here, we consider such transitions under the addition of local random white noise and measurement inefficiency in an XX spin chain. We identify a nonmonotonic dependence on the local noise strength in both the averaged entanglement and operator correlations, specifically the subsystem parity variance. While the nonmonotonicity persists at any finite efficiency for the operator correlations, it disappears at finite inefficiency for the entanglement. The analysis of scaling with the system size in a finite-length chain indicates that, at finite efficiency, this effect leads to distinct measurement-induced phase transitions for operator correlations and entanglement. Our result hints at a difference between area-law entanglement scaling and Zeno-localized phases for inefficient monitoring.