We examine the kinematics of the gas in the environments of galaxies hosting quasars at z~2. We employ 148 projected quasar pairs to study the circumgalactic gas of the foreground quasars in absorption. The sample selects foreground quasars with precise redshift measurements, using emission lines with precision <~300km/s and average offsets from the systemic redshift <~|100km/s|. We stack the background quasar spectra at the foreground quasar's systemic redshift to study the mean absorption in C II, C IV, and Mg II. We find that the mean absorptions exhibit large velocity widths {sigma}_v_~300km/s. Further, the mean absorptions appear to be asymmetric about the systemic redshifts. The mean absorption centroids exhibit small redshift relative to the systemic {delta}v~+200km/s, with large intrinsic scatter in the centroid velocities of the individual absorption systems. We find the observed widths are consistent with gas in gravitational motion and Hubble flow. However, while the observation of large widths alone does not require galactic-scale outflows, the observed offsets suggest that the gas is on average outflowing from the galaxy. The observed offsets also suggest that the ionizing radiation from the foreground quasars is anisotropic and/or intermittent.