  Non-Hermitian dynamics is ubiquitous in various physical systems. While
recent study shows that such a dynamics leads to an area-law scaling of the
entanglement entropy due to the non-Hermitian skin effects, it remains unclear
how disorder changes the behavior of the entanglement entropy in a
non-Hermitian system with skin effects. Here we study the dynamics of a
many-body state of free fermions in the paradigmatic Hatano-Nelson model with
open boundaries, and find that the area-law behavior of the entanglement
entropy in the pristine Hatano-Nelson model develops into a logarithmic scaling
for small disorder strength. As we further increase the disorder strength, the
system reenters an area-law regime through an entanglement phase transition. At
the critical point, the entanglement entropy exhibits a universal algebraic
scaling. We further demonstrate the absence of a conformal invariance in the
log-law regime by examining the subsystem entanglement entropy, the connected
correlation function and the mutual information. Finally, we show the existence
of disorder induced entanglement phase transitions in the Hatano-Nelson model
with periodic boundaries.
