  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 a XX spin chain. We identify a non-monotonic dependence on the
local noise strength in both the averaged entanglement and operator
correlations, specifically the subsystem parity variance. While the
non-monotonicity 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 MiPTs for operator
correlations and entanglement. Our result hints at a difference between
area-law entanglement scaling and Zeno-localized phases for inefficient
monitoring.
