We report the multiwavelength properties of millimeter galaxies hosting X-ray detected active galactic nuclei (AGNs) from the ALMA Lensing Cluster Survey (ALCS). ALCS is an extensive survey of well-studied lensing clusters with ALMA, covering an area of 133 arcmin(2) over 33 clusters with a 1.2 mm flux-density limit of similar to 60 mu Jy (1 sigma). Utilizing the archival data of Chandra, we identify three AGNs at z = 1.06, 2.09, and 2.84 among the 180 millimeter sources securely detected in the ALCS (of which 155 are inside the coverage of Chandra). The X-ray spectral analysis shows that two AGNs are not significantly absorbed (log N-H cm (2) < 23 -), while the other shows signs of moderate absorption (log NH cm (2) similar to 23.5 -). We also perform spectral energy distribution modeling of X-ray to millimeter photometry. We find that our X-ray AGN sample shows both high mass-accretion rates (intrinsic 0.5-8 keV X-ray luminosities of similar to 10(44-45) erg s(-1)) and star formation rates (> 100 Me yr-1). This demonstrates that a wide-area survey with ALMA and Chandra can selectively detect intense growth of both galaxies and supermassive black holes in the high-redshift universe.