  The non-Hermitian skin effect, i.e. eigenstate condensation at the edges in
lattices with open boundaries, is an exotic manifestation of non-Hermitian
systems. In Bloch theory, an effective non-Hermitian Hamiltonian is generally
used to describe dissipation, which however is not norm-preserving and neglects
quantum jumps. Here it is shown that in a self-consistent description of the
dissipative dynamics in a one-band lattice, based on the stochastic
Schr\"odinger equation or Lindblad master equation with a collective jump
operator, the skin effect and its dynamical features are washed out.
Nevertheless, both short- and long-time relaxation dynamics provide a hidden
signature of the skin effect found in the semiclassical limit. In particular,
relaxation toward a maximally mixed state with the largest von Neumann entropy
in a lattice with open boundaries is a manifestation of the semiclassical skin
effect.
