  We investigate the dynamics of the fermionic logarithmic negativity in a
free-fermion chain with a localized loss, which acts as a dissipative impurity.
The chain is initially prepared in a generic Fermi sea. In the standard
hydrodynamic limit of large subsystems and long times, with their ratio fixed,
the negativity between two subsystems is described by a simple formula, which
depends only on the effective absorption coefficient of the impurity. The
negativity grows linearly at short times, then saturating to a volume-law
scaling. Physically, this reflects the continuous production with time of
entangling pairs of excitations at the impurity site. Interestingly, the
negativity is not the same as the R\'enyi mutual information with R\'enyi index
$1/2$, in contrast with the case of unitary dynamics. This reflects the
interplay between dissipative and unitary processes. The negativity content of
the entangling pairs is obtained in terms of an effective two-state mixed
density matrix for the subsystems. Criticality in the initial Fermi sea is
reflected in the presence of logarithmic corrections. The prefactor of the
logarithmic scaling depends on the loss rate, suggesting a nontrivial interplay
between dissipation and criticality.
