  Monitored quantum dynamics -- unitary evolution interspersed with
measurements -- has recently emerged as a rich domain for phase structure in
quantum many-body systems away from equilibrium. Here we study monitored
dynamics from the point of view of an eavesdropper who has access to the
classical measurement outcomes, but not to the quantum many-body system. We
show that a measure of information flow from the quantum system to the
classical measurement record -- the informational power -- undergoes a phase
transition in correspondence with the measurement-induced phase transition
(MIPT). This transition determines the eavesdropper's (in)ability to learn
properties of an unknown initial quantum state of the system, given a complete
classical description of the monitored dynamics and arbitrary classical
computational resources. We make this learnability transition concrete by
defining classical shadows protocols that the eavesdropper may apply to this
problem, and show that the MIPT manifests as a transition in the sample
complexity of various shadow estimation tasks, which become harder in the
low-measurement phase. We focus on three applications of interest: Pauli
expectation values (where we find the MIPT appears as a point of optimal
learnability for typical Pauli operators), many-body fidelity, and global
charge in $U(1)$-symmetric dynamics. Our work unifies different manifestations
of the MIPT under the umbrella of learnability and gives this notion a general
operational meaning via classical shadows.
