  The non-Hermitian skin effect (NHSE), which describes the localization of
macroscopic fraction of eigenstates at a specific boundary, is inherently tied
to nonreciprocity. Here, we show that the NHSE can be engineered in an open
magnetic system interacting with a reciprocal reservoir, through the interplay
between the reservoir-induced coherent and dissipative couplings. Based on a
Lindbladian time evolution, we investigate the transient nonreciprocal dynamics
along a spin chain and its unidirectional limit, which allow us to reveal both
consistency with and limitation of the non-Hermitian Hamiltonian approach. We
comment on the connection to the semiclassical dissipative magnetization
dynamics, and identify the key ingredients underlying the NHSE in magnetic
systems as Dzyaloshinskii-Moriya interaction (DMI) and dissipative coupling,
analogous to those in multi-layered magnet-metal heterostructures. Our work
suggests the generality of nonreciprocal dynamics in magnetic systems and may
inspire new schemes for engineering nonreciprocity in other quantum platforms.
