  We study a one-dimensional lattice system of free fermions subjected to a
generalized measurement process: the system exchanges particles with its
environment, but each fermion leaving or entering the system is counted. In
contrast to the freezing of dynamics due to frequent measurements of
lattice-site occupation numbers, a high rate of fermion counts induces fast
fluctuations in the state of the system. Still, through numerical simulations
of quantum trajectories and an analytical approach based on replica Keldysh
field theory, we find that instantaneous correlations and entanglement
properties of free fermions subjected to fermion counting and local occupation
measurements are strikingly similar. We explain this similarity through a
generalized Zeno effect induced by fermion counting and a universal
long-wavelength description in terms of an $\mathrm{SU}(R)$ nonlinear sigma
model. Further, for both types of measurement processes, we present strong
evidence against the existence of a critical phase with logarithmic
entanglement and conformal invariance at finite measurement rates. Instead, we
identify a well-defined and finite critical range of length scales on which
signatures of conformal invariance are observable. While area-law entanglement
is established beyond a scale that is exponentially large in the measurement
rate, the upper boundary of the critical range is only algebraically large and
thus numerically accessible.
