We use new central stellar velocity dispersions and nuclear X-ray and Halpha luminosities for the Palomar survey of nearby galaxies to investigate the distribution of nuclear bolometric luminosities and Eddington ratios for their central black holes (BHs). This information helps to constrain the nature of their accretion flows and the physical drivers that control the spectral diversity of nearby active galactic nuclei. The characteristic values of the bolometric luminosities and Eddington ratios, which span over 7-8 orders of magnitude, from L_bol < 10^37 to 3 X 10^44 erg/s and L_bol/L_Edd ~ 10^-9 to 10^-1, vary systematically with nuclear spectral classification, increasing along the sequence absorption-line nuclei --> transition objects --> LINERs --> Seyferts. The Eddington ratio also increases from early-type to late-type galaxies. We show that the very modest accretion rates inferred from the nuclear luminosities can be readily supplied through local mass loss from evolved stars and Bondi accretion of hot gas, without appealing to additional fueling mechanisms such as angular momentum transport on larger scales. Indeed, we argue that the fuel reservoir generated by local processes should produce far more active nuclei than is actually observed. This generic luminosity-deficit problem suggests that the radiative efficiency in these systems is much less than the canonical value of 0.1 for traditional optically thick, geometrically thin accretion disks. The observed values of L_bol/L_Edd, all substantially below unity, further support the hypothesis that massive BHs in most nearby galaxies reside in a low or quiescent state, sustained by accretion through a radiatively inefficient mode.