We observed B1600+434 and B1608+656 with CXO/ACIS, detecting both quasar images in B1600+434 and three of four images in B1608+656. We did not detect significant X-ray emission from nearby galaxy groups or clusters associated with each lens galaxy. The upper limits on the X-ray luminosity of any cluster within 4 arcmin of each lens and at each lens redshift are of ~2\times10^{42} and ~6\times10^{42} erg/s for B1600+434 and B1608+656, respectively. The radio-loud source quasars have power-law photon indices of Γ=1.9±0.2 and Γ=1.4±0.3 and X-ray luminosities of 1.4^{+0.2}_{-0.1}\times10^{45} and 2.9^{+0.7}_{-0.4}\times10^{44} erg/s for B1600+434 and B1608+656, respectively before correcting for the magnification. We detected a differential absorption column density of Δ N_H ~ 3\times10^{21} cm^{-2} between the two images of B1600+434, roughly consistent with expectations from differential extinction estimates of Δ E(B-V)=0.1 mag and a standard dust-to-gas ratio. The differential absorption observed in gravitational lenses may serve as an important probe to study the gas content in high redshift galaxies since it can separate the absorbing column originating from the lens galaxy and those intrinsic to quasars. We also detected 157 serendipitous X-ray sources in the two Chandra fields and identified the brighter optical counterparts using the SDSS and DPOSS surveys. ; Comment: 24 pages, including 3 figures and 5 tables, accepted by ApJ