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.
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