We have applied optical absorption spectroscopy to investigate van der Waals heterostructures formed of pairs of monolayer transition metal dichalcogenide crystals, choosing MoS₂/WS₂ and MoSe₂/WSe₂ as test cases. In the heterostructure spectra, we observe a significant broadening of the excitonic transitions compared to the corresponding features in the isolated layers. The broadening is interpreted as a lifetime effect arising from decay of excitons initially created in either layer through charge transfer processes expected for a staggered band alignment. The measured spectral broadening of 20 meV-35 meV implies lifetimes for charge separation of the near band-edge A and B excitons in the range of 20-35 fs. Higher-lying transitions exhibit still greater broadening.