  The conducting state of the quasi-two-dimensional organic conductor,
$\alpha$-(BEDT-TTF)$_2$I$_3$, at ambient pressure is investigated with $^{13}$C
NMR measurements, which separate the local electronic states at three
nonequivalent molecular sites (A, B, and C). The spin susceptibility and
electron correlation effect are revealed in a locally resolved manner. While
there is no remarkable site-dependence around room temperature, the local spin
susceptibility gradually disproportionates among the nonequivalent sites with
decreasing temperature. The disproportionation-ratio yields 5:4:6 for A:B:C
molecules at 140 K. Distinct site- and temperature-dependences are also
observed in the Korringa ratio, $\mathcal{K}_i \propto (1/T_{1}T)_iK^{-2}_i$
($i$ = A, B, and C), which is a measure of the strength and the type of
electron correlations. The values of $\mathcal{K}_i$ point to sizable
antiferromagnetic spin correlation. We argue the present results in terms of
the theoretical prediction of the peculiar site-specific reciprocal-space
($\bm{k}$-space) anisotropy on the tilted Dirac cone, and discuss the
$\bm{k}$-dependent profiles of the spin susceptibility and electron correlation
on the cone.
