  We develop a method to calculate left-right eigenvector correlations of the
product of $m$ independent $N\times N$ complex Ginibre matrices. For
illustration, we present explicit analytical results for the vector overlap for
a couple of examples for small $m$ and $N$. We conjecture that the integrated
overlap between left and right eigenvectors is given by the formula $O = 1 +
(m/2)(N-1)$ and support this conjecture by analytical and numerical
calculations. We derive an analytical expression for the limiting correlation
density as $N\rightarrow \infty$ for the product of Ginibre matrices as well as
for the product of elliptic matrices. In the latter case, we find that the
correlation function is independent of the eccentricities of the elliptic laws.
