  We study the competing effects of collective generalized measurements and
interaction-induced scrambling in the dynamics of an ensemble of spin-1/2
particles at the level of quantum trajectories. This setup can be considered as
analogous to the one leading to measurement-induced transitions in quantum
circuits. We show that the interplay between collective unitary dynamics and
measurements leads to three regimes of the average Quantum Fisher Information
(QFI), which is a witness of multipartite entanglement, as a function of the
monitoring strength. While both weak and strong measurements lead to extensive
QFI density (i.e., individual quantum trajectories yield states displaying
Heisenberg scaling), an intermediate regime of classical-like states emerges
for all system sizes where the measurement effectively competes with the
scrambling dynamics and precludes the development of quantum correlations,
leading to sub-Heisenberg-limited states. We characterize these regimes and the
crossovers between them using numerical and analytical tools, and discuss the
connections between our findings, entanglement phases in monitored many-body
systems, and the quantum-to-classical transition.
