Consider the problem of storing data in a distributed manner over $T$ servers. Specifically, the data needs to (i) be recoverable from any $τ$ servers, and (ii) remain private from any $z$ colluding servers, where privacy is quantified in terms of mutual information between the data and all the information available at any $z$ colluding servers. For this model, our main results are (i) the fundamental trade-off between storage size and the level of desired privacy, and (ii) the optimal amount of local randomness necessary at the encoder. As a byproduct, our results provide an optimal lower bound on the individual share size of ramp secret sharing schemes under a more general leakage symmetry condition than the ones previously considered in the literature.