  We consider a one-dimensional system of non-interacting fermions featuring
both boundary driving and continuous monitoring of the bulk particle density.
Due to the measurements, the expectation values of the local density and
current operators are random variables whose average behavior is described by a
well studied Lindblad master equation. By means of exact numerical
computations, we go beyond the averaged dynamics and study their full
probability distribution functions, focusing on the late-time stationary
regime. We find that, contrary to the averaged values, the spatial profiles of
the median density and current are non-trivial, exhibiting qualitative
differences as a function of the monitoring strength. At weak monitoring, the
medians are close to the means, displaying diffusive spatial profiles. At
strong monitoring, we find that the median density and current develop a
domain-wall and single-peak profile, respectively, which are suggestive of a
Zeno-like localization in typical quantum trajectories. While we are not able
to identify a sharp phase transition as a function of the monitoring rate, our
work highlights the usefulness of characterizing typical behavior beyond the
averaged values in the context of monitored many-body quantum dynamics.
