  We consider five indicators for intrinsic AGN luminosity: the luminosities of
the [OIII]$\lambda$5007 line, the [OIV]25.89$\mu$m line, the mid-infrared (MIR)
continuum emission by the torus, and the radio and hard X-ray (E $>$ 10keV)
continuum emission. We compare these different proxies using two complete
samples of low-redshift type 2 AGN selected in a homogeneous way based on
different indicators: an optically selected [OIII] sample and a mid-infrared
selected 12$\mu$m sample. We examine the correlations between all five
different proxies, and find better agreement for the [OIV], MIR, and [OIII]
luminosities than for the hard X-ray and radio luminosities. Next, we compare
the ratios of the fluxes of the different proxies to their values in unobscured
Type 1 AGN. The agreement is best for the ratio of the [OIV] and MIR fluxes,
while the ratios of the hard X-ray to [OIII], [OIV], and MIR fluxes are
systematically low by about an order-of-magnitude in the Type 2 AGN, indicating
that hard X-ray selected samples do not represent the full Type 2 AGN
population. In a similar spirit, we compare different optical and MIR
diagnostics of the relative energetic contributions of AGN and star formation
processes in our samples of Type 2 AGN. We find good agreement between the
various diagnostic parameters, such as the equivalent width of the MIR
polycyclic aromatic hydrocarbon features, the ratio of the MIR [OIV]/[NeII]
emission-lines, the spectral index of the MIR continuum, and the commonly used
optical emission-line ratios. Finally, we test whether the presence of cold gas
associated with star-formation leads to an enhanced conversion efficiency of
AGN ionizing radiation into [OIII] or [OIV] emission. We find that no
compelling evidence exists for this scenario for the luminosities represented
in this sample (L$_{bol}$ $\approx$ 10$^{9}$ - 8 $\times$ 10$^{11}$
L$_{\sun}$). (abridged)
