  We derive laws for the distribution of quantum steering among different
parties in multipartite Gaussian states under Gaussian measurements. We prove
that a monogamy relation akin to the generalized Coffman-Kundu-Wootters
inequality holds quantitatively for a recently introduced measure of Gaussian
steering. We then define the residual Gaussian steering, stemming from the
monogamy inequality, as an indicator of collective steering-type correlations.
For pure three-mode Gaussian states, the residual acts a quantifier of genuine
multipartite steering, and is interpreted operationally in terms of the
guaranteed key rate in the task of secure quantum secret sharing. Optimal
resource states for the latter protocol are identified, and their possible
experimental implementation discussed. Our results pin down the role of
multipartite steering for quantum communication.
