  Computationally efficient matrix multiplication is a fundamental requirement
in various fields, including and particularly in data analytics. To do so, the
computation task of a large-scale matrix multiplication is typically outsourced
to multiple servers. However, due to data misusage at the servers, security is
typically of concern. In this paper, we study the two-sided secure matrix
multiplication problem, where a user is interested in the matrix product
$\boldsymbol{AB}$ of two finite field private matrices $\boldsymbol{A}$ and
$\boldsymbol{B}$ from an information-theoretic perspective. In this problem,
the user exploits the computational resources of $N$ servers to compute the
matrix product, but simultaneously tries to conceal the private matrices from
the servers. Our goal is twofold: (i) to maximize the communication rate, and,
(ii) to minimize the effective number of server observations needed to
determine $\boldsymbol{AB}$, while preserving security, where we allow for up
to $\ell\leq N$ servers to collude. To this end, we propose a general aligned
secret sharing scheme for which we optimize the matrix partition of matrices
$\boldsymbol{A}$ and $\boldsymbol{B}$ in order to either optimize objective (i)
or (ii) as a function of the system parameters (e.g., $N$ and $\ell$). A
proposed inductive approach gives us analytical, close-to-optimal solutions for
both (i) and (ii). With respect to (i), our scheme significantly outperforms
the existing scheme of Chang and Tandon in terms of (a) communication rate, (b)
maximum tolerable number of colluding servers and (c) computational complexity.
