We study the trade-off between communication rate and privacy for distributed batch matrix multiplication of two independent sequences of matrices $\mathbf{A}$ and $\mathbf{B}$ with uniformly distributed entries. In our setting, $\mathbf{B}$ is publicly accessible by all the servers while $\mathbf{A}$ must remain private. A user is interested in evaluating the product $\mathbf{AB}$ with the responses from the $k$ fastest servers. For a given parameter $α\in [0, 1]$, our privacy constraint must ensure that any set of $\ell$ colluding servers cannot learn more than a fraction $α$ of $\mathbf{A}$. Additionally, we study the trade-off between the amount of local randomness needed at the encoder and privacy. Finally, we establish the optimal trade-offs when the matrices are square and identify a linear relationship between information leakage and communication rate.