  An algebra $A$ is said to be two-sided zero product determined if every
bilinear functional $\varphi:A\times A\to F$ satisfying $ \varphi(x,y)=0$
whenever $xy=yx=0$ is of the form $\varphi(x,y)=\tau_1(xy) + \tau_2(yx)$ for
some linear functionals $\tau_1,\tau_2$ on $A$. We present some basic
properties and equivalent definitions, examine connections with some properties
of derivations, and as the main result prove that a finite-dimensional simple
algebra that is not a division algebra is two-sided zero product determined if
and only if it is separable.
