We present the results from a Chandra pilot study of 12 massive galaxy mergers selected from Galaxy Zoo. The sample includes major mergers down to a host galaxy mass of 1011 M ☉ that already have optical active galactic nucleus (AGN) signatures in at least one of the progenitors. We find that the coincidences of optically selected active nuclei with mildly obscured (N H 1.1 × 1022 cm–2) X-ray nuclei are relatively common (8/12), but the detections are too faint (<40 counts per nucleus; f 2-10 keV 1.2 × 10–13 erg s–1 cm–2) to reliably separate starburst and nuclear activity as the origin of the X-ray emission. Only one merger is found to have confirmed binary X-ray nuclei, though the X-ray emission from its southern nucleus could be due solely to star formation. Thus, the occurrences of binary AGNs in these mergers are rare (0%-8%), unless most merger-induced active nuclei are very heavily obscured or Compton thick.