We present new observations of CO J = 2 - 1 emission from the protoplanetary disk around TW Hya. Emission is detected out to 240 au (4″) and found to exhibit azimuthal variations up to 20% beyond 180 au (3″), with the west side of the disk brighter than the east. This asymmetry is interpreted as tracing the shadow previously seen in scattered light. A re-analysis of the multi-epoch observations of the dust shadow in scattered light from Debes et al. suggests that an oscillatory motion would provide a better model of the temporal evolution of the dust shadow rather than orbital motion. Both models predict an angular offset between the dust shadow and the gas shadow of up to ~100°. We attribute this offset to the finite rate at which dust grains and gas molecules can exchange heat, dominated by the collisional rate between gas molecules and dust grains, t _coll. The angular offsets derived are equivalent to collisional timescales that range from the near-instantaneous up to t _coll ~ 10 yr, depending on whether a straight or curved dust shadow, as suggested by Hubble Space Telescope observations reported by Debes et al., is adopted. The inferred range of t _coll are consistent with those predictions based on representative gas densities, temperatures, gas-to-dust ratios and grain sizes. These results represent the first time empirical constraints can be placed on t _coll.