  We explore anomalous skin effects at non-Hermitian impurities by studying
their interplay with potential disorder and by exactly solving a minimal
lattice model. A striking feature of the solvable single-impurity model is that
the presence of anisotropic hopping terms can induce a scale-free accumulation
of all eigenstates opposite to the bulk hopping direction, although the
nonmonotonic behavior is fine tuned and further increasing such hopping weakens
and eventually reverses the effect. The interplay with bulk potential disorder,
however, qualitatively enriches this phenomenology leading to a robust
nonmonotonic localization behavior as directional hopping strengths are tuned.
Nonmonotonicity persists even in the limit of an entirely Hermitian bulk with a
single non-Hermitian impurity.
