  In this paper, we study the problem of distributed multi-user secret sharing,
including a trusted master node, $N\in \mathbb{N}$ storage nodes, and $K$
users, where each user has access to the contents of a subset of storage nodes.
Each user has an independent secret message with certain rate, defined as the
size of the message normalized by the size of a storage node. Having access to
the secret messages, the trusted master node places encoded shares in the
storage nodes, such that (i) each user can recover its own message from the
content of the storage nodes that it has access to, (ii) each user cannot gain
any information about the message of any other user. We characterize the
capacity region of the distributed multi-user secret sharing, defined as the
set of all achievable rate tuples, subject to the correctness and privacy
constraints. In the achievable scheme, for each user, the master node forms a
polynomial with the degree equal to the number of its accessible storage nodes
minus one, where the value of this polynomial at certain points are stored as
the encoded shares. The message of that user is embedded in some of the
coefficients of the polynomial. The remaining coefficients are determined such
that the content of each storage node serves as the encoded shares for all
users that have access to that storage node.
