Exciton transfer between two parallel GaAs V-groove quantum wires or two planar quantum wells separated by AlGaAs barriers ranging from 5.5 nm to 20 nm thickness is studied by photoluminescence and photoluminescence excitation spectroscopy. It is found that the transfer is strongly reduced between the widely spaced quantum wires as compared with quantum wells. This observation is supported by model calculations, which yield strong dimensionality dependence of the photon-exchange transfer