  Distribution of quantum correlations among remote users is a key procedure
underlying many quantum information technologies. Einstein-Podolsky-Rosen
steering, which is one kind of such correlations stronger than entanglement,
has been identified as a resource for secure quantum networks. We show that
this resource can be established between two and even more distant parties by
transmission of a system being separable from all the parties. For the case
with two parties, we design a protocol allowing to distribute one-way Gaussian
steering between them via a separable carrier; the obtained steering can be
used subsequently for one-sided device-independent (1sDI) quantum key
distribution. Further, we extend the protocol to three parties, a scenario
which exhibits richer steerability properties including one-to-multimode
steering and collective steering, and which can be used for 1sDI quantum secret
sharing. All the proposed steering distribution protocols can be implemented
with squeezed states, beam splitters and displacements, and thus they can be
readily realized experimentally. Our findings reveal that not only entanglement
but even steering can be distributed via communication of a separable system.
Viewed from a different perspective, the present protocols also demonstrate
that one can switch multipartite states between different steerability classes
by operations on parts of the states.
