Is Mg II 2800 a Reliable Virial Broadening Estimator for Quasars?

[Abridged] Broad MgII 2800 and Hbeta lines have emerged as the most reliable virial estimators of black hole mass in quasars. Which is more reliable? Comparison of MgII 2800 and Hbeta profile measures in the same sources and especially FWHM measures that provide the virial broadening estimator. Identification of 680 bright Sloan Digital Sky Survey DataRelease 7 quasars with spectra showing both MgII 2800 and Hbeta lines, at redshift 0.4 < z < 0.75. The s/n of these spectra are high enough to allow binning in the "four-dimensional (4D) eigenvector 1'' optical plane and construction of high s/n composite spectra. We confirm that MgII 2800 shows a profile that is ~ 20% narrower as suggested in some previous studies. FWHM measures for Population B sources (i.e., with FWHM of Hbeta larger than 4000 km/s) are uncertain because they show complex profiles with at least two broad-line components involving a nearly unshifted broad and redshifted very-broad component. Only the broad component is likely to be a valid virial estimator. If Hbeta and MgII 2800 are not corrected for the very broad component then black hole mass values for Population B sources will be systematically overestimated by up to logM ~ 0.3-0.4 dex. We suggest a simple correction that can be applied to the majority of sources. MgII 2800 is the safer virial estimator for Population B sources because the centroid shifts with respect to rest frame are lower than for Hbeta. In the broad and very broad component profile interpretation this is a consequence of the lower very broad to broad component intensity ratio for MgII 2800. Effective discrimination of black hole mass and Eddington ratio at fixed redshift is not achieved via luminosity binning but rather by binning in a "4D eigenvector 1'' context that reflects different broad line region geometry/kinematics likely driven by Eddington ratio.