We present new Keck/MOSFIRE K-band spectroscopy for a sample of 14 faint, X-ray-selected active galactic nuclei (AGNs) in the COSMOS field. The data cover the spectral region surrounding the broad Balmer emission lines, which enables the estimation of black hole masses (MBH) and accretion rates (in terms of L LEdd). We focus on 10 AGNs at z ; 3.3, where we observe the Hβ spectral region, while for the other four z ; 2.4 sources we use the Ha broad emission line. Compared with previous detailed studies of unobscured AGNs at these high redshifts, our sources are fainter by an order of magnitude, corresponding to number densities of order ∼10−6 –10−5 M - pc 3. The lower AGN luminosities also allow for a robust identification of the host galaxy emission, necessary to obtain reliable intrinsic AGN luminosities, BH masses and accretion rates. We find the AGNs in our sample to be powered by supermassive black holes (SMBHs) with a typical mass of M M BH  5 10 ´  8 —significantly lower than the higher-luminosity, rarer quasars reported in earlier studies. The accretion rates are in the range L LEdd ∼ 0.1–0.4, with an evident lack of sources with lower L LEdd (and higher MBH), as found in several studies of faint AGNs at intermediate redshifts. Based on the early growth expected for the SMBHs in our sample, we argue that a significant population of faint z ∼ 5−6 AGNs, with M M BH ~ 10  6 , should be detectable in the deepest X-ray surveys available, but this is not observed. We discuss several possible explanations for the apparent absence of such a population, concluding that the most probable scenario involves an evolution in source obscuration and/or radiative efficiencies.