  We identify a phase transition between two kinds of volume-law entangled
phases in non-local but few-body unitary dynamics with local projective
measurements. In one phase, a finite fraction of the system belongs to a
fully-entangled state, one for which no subsystem is in a pure state, while in
the second phase, the steady-state is a product state over extensively many,
finite subsystems. We study this "separability" transition in a family of
solvable models in which we analytically determine the transition point, the
evolution of certain entanglement properties of interest, and relate this to a
mean-field percolation transition. Since the entanglement entropy density does
not distinguish these phases, we introduce the entangling power - which
measures whether local measurements outside of two finite subsystems can boost
their mutual information - as an order parameter, after considering its
behavior in tensor network states, and numerically studying its behavior in a
model of Clifford dynamics with measurements. We argue that in our models, the
separability transition coincides with a transition in the computational
"hardness" of classically determining the output probability distribution for
the steady-state in a certain basis of product states. A prediction for this
distribution, which is accurate in the separable phase, and should deviate from
the true distribution in the fully-entangled phase, provides a possible
benchmarking task for quantum computers.
