We present Chandra X-ray imaging and spectroscopy for 14 quasars in spatially resolved pairs, part of a complete sample of binary quasars with small transverse separations drawn from Sloan Digital Sky Survey (DR6) photometry. We find no significant difference in X-ray properties when compared with large control samples of isolated quasars. We present infrared photometry from our observations with SWIRC at the MMT, and from the WISE Preliminary Data Release, and fit simple spectral energy distributions to all 14 QSOs. We find preliminary evidence that substantial contributions from star formation are required, but possibly no more so than for isolated X-ray-detected QSOs. Sensitive searches of the X-ray images for extended emission, and the optical images for optical galaxy excess show that these binary QSOs are not preferentially found in rich cluster environments. While larger binary QSO samples with richer far-IR and sub-millimeter multiwavelength data might better reveal signatures of merging and triggering, optical color-selection of QSO pairs may be biased against such signatures. X-ray and/or variability selection of QSO pairs, while challenging, should be attempted. We present in our Appendix a primer on X-ray flux and luminosity calculations.