We consider the problem of storing data in a distributed manner over T servers. We require the data (i) to be recoverable from the T servers, and (ii) to remain private from any T -1 colluding servers, where privacy is quantified in terms of mutual information between the data and all the information available at the T -1 colluding servers. For this model, we determine (i) the fundamental trade-off between storage size and the level of desired privacy, (ii) the optimal amount of local randomness necessary at the encoder, and (iii) an explicit low-complexity coding scheme that solely relies on XOR operations and that asymptotically (with the data size) matches the fundamental limits found.