  Recently, there has been interest in the dynamics of monitored quantum
systems using linear jump operators related to the creation or annihilation of
particles. Here we study the dynamics of the entanglement entropy under quantum
jumps that induce local particle losses in a model of free fermions with
hopping and $\mathbb{Z}_2$ pairing. We explore the different steady-state
entanglement regimes by interpolating between monitored free fermions with U(1)
symmetry and $\mathbb{Z}_2$ fermions. In the absence of pairing, the U(1)
symmetric model approaches the vacuum at long times, with the entanglement
entropy showing non-monotonic behavior over time that we capture with a
phenomenological quasiparticle ansatz. In this regime, quantum jumps play a key
role, and we highlight this by exactly computing their waiting-time
distribution. On the other hand, the interplay between losses and pairing in
the $\mathbb{Z}_2$ case gives rise to quantum trajectories with entangled
steady-states. We show that by tuning the several system parameters, a
measurement-induced entanglement transition occurs where the entanglement
entropy scaling changes from logarithmic to area-law. We compare this
transition with the one derived in the no-click limit and observe qualitative
agreement in most of the phase diagram. Furthermore, the statistics of
entanglement gain and loss are analyzed to better understand the impact of the
linear jump operators.
