Deep SCUBA surveys have uncovered a large population of massive submillimeter-emitting galaxies (SMGs; f_850μm 4 mJy) at z 1. Although it is generally believed that these galaxies host intense star formation activity, there is growing evidence that a substantial fraction also harbor an active galactic nucleus (AGN; i.e., an accreting super-massive black hole [SMBH]). We present here possibly the strongest evidence for this viewpoint to date: the combination of ultradeep X-ray observations (the 2 Ms Chandra Deep Field-North) and deep Keck spectroscopic data of SMGs with radio counterparts. We find that the majority (≈75%) of these radio-selected spectroscopically identified SMGs host AGN activity; the other ≈25% have X-ray properties consistent with star formation (X-ray-derived star formation rates of ≈1300-2700 M yr^-1). The AGNs have properties generally consistent with those of nearby luminous AGNs (Γ ≈ 1.8 ± 0.5, N_H ≈ 10²⁰-10²⁴ cm^-2, and L_X ≈ 10⁴³-10^44.5 ergs s^-1), and the majority (≈80%) are heavily obscured (N_H 10²³ cm^-2). We construct composite rest-frame 2-20 keV spectra for three different obscuration classes [N_H < 10²³ cm^-2, N_H = (1-5) × 10²³ cm^-2, and N_H > 5 × 10²³ cm^-2], which reveal features not seen in the individual X-ray spectra. An ≈1 keV equivalent width Fe Kα emission line is seen in the composite X-ray spectrum of the most heavily obscured AGNs, suggesting Compton-thick or near Compton-thick absorption. Even taking into account the effects of absorption, we find that the average X-ray to far-IR luminosity ratio of the AGN-classified SMGs (L_X/L_FIR = 0.004) is approximately 1 order of magnitude below that found for typical quasars. This result suggests that intense star formation activity (of order ≈1000 M yr^-1) dominates the bolometric output of these SMGs. However, we also explore the possibility that the X-ray to far-IR luminosity ratio of the AGN components is intrinsically less than that found for typical quasars and postulate that some SMGs may be AGN dominated. We investigate the implications of our results for the growth of massive black holes, discuss the prospects for deeper X-ray observations, and explore the scientific potential offered by the next generation of X-ray observatories.