The resilience of quantum entanglement to a classicality-inducing environment is tied to fundamental aspects of quantum many-body systems. The dynamics of entanglement has recently been studied in the context of measurement-induced entanglement transitions, where the steady-state entanglement collapses from a volume-law to an area-law at a critical measurement probability $p_{c}$. Interestingly, there is a distinction in the value of $p_{c}$ depending on how well the underlying unitary dynamics scramble quantum information. For strongly chaotic systems, $p_{c} > 0$, whereas for weakly chaotic systems, such as integrable models, $p_{c} = 0$. In this work, we investigate these measurement-induced entanglement transitions in a system where the underlying unitary dynamics are many-body localized (MBL). We demonstrate that the emergent integrability in an MBL system implies a qualitative difference in the nature of the measurement-induced transition depending on the measurement basis, with $p_{c} > 0$ when the measurement basis is scrambled and $p_{c} = 0$ when it is not. This feature is not found in Haar-random circuit models, where all local operators are scrambled in time. When the transition occurs at $p_{c} > 0$, we use finite-size scaling to obtain the critical exponent $ u = 1.3(2)$, close to the value for 2+0D percolation. We also find a dynamical critical exponent of $z = 0.98(4)$ and logarithmic scaling of the R{e}nyi entropies at criticality, suggesting an underlying conformal symmetry at the critical point. This work further demonstrates how the nature of the measurement-induced entanglement transition depends on the scrambling nature of the underlying unitary dynamics. This leads to further questions on the control and simulation of entangled quantum states by measurements in open quantum systems.
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