We present the analysis and results of a spectroscopic follow-up program of a mass-selected sample of six galaxies at 3< z< 4 using data from Keck-NIRPSEC and VLT-Xshooter. We confirm the z> 3 redshifts for half of the sample through the detection of strong nebular emission lines, and improve the z phot accuracy for the remainder of the sample through the combination of photometry and spectra. The modeling of the emission-line-corrected spectral energy distributions (SEDs) adopting improved redshifts confirms the very large stellar masses of the sample ({M}* ∼ 1.5{--}4× {10}11{M}ȯ ) in the first 2 Gyr of cosmic history, with a diverse range in stellar ages, star-formation rates, and dust content. From the analysis of emission-line luminosities and widths, and far-infrared (FIR) fluxes, we confirm that ≳ 80 % of the sample are hosts to luminous hidden active galactic nuclei (AGNs), with bolometric luminosities of ∼1044–46 erg s‑1. We find that the MIPS 24 μm photometry is largely contaminated by AGN continuum, rendering the SFRs derived using only 24 μm photometry to be severely overestimated. By including the emission from the AGN in the modeling of the UV-to-FIR SEDs, we confirm that the presence of the AGN does not considerably bias the stellar masses (< 0.3 dex at 1σ). We show evidence for a rapid increase of the AGN fraction from ∼30% to ∼60%–100% over the 1 Gyr between z∼ 2 and z∼ 3. Although we cannot exclude some enhancement of the AGN fraction for our sample due to selection effects, the small measured [O iii] contamination to the observed K-band fluxes suggests that our sample is not significantly biased toward massive galaxies hosting AGNs.