A joint determination of the reactor antineutrino spectra resulting from the fission of ^{235}U and ^{239}Pu has been carried out by the Daya Bay and PROSPECT Collaborations. This Letter reports the level of consistency of ^{235}U spectrum measurements from the two experiments and presents new results from a joint analysis of both data sets. The measurements are found to be consistent. The combined analysis reduces the degeneracy between the dominant ^{235}U and ^{239}Pu isotopes and improves the uncertainty of the ^{235}U spectral shape to about 3%. The ^{235}U and ^{239}Pu antineutrino energy spectra are unfolded from the jointly deconvolved reactor spectra using the Wiener-SVD unfolding method, providing a data-based reference for other reactor antineutrino experiments and other applications. This is the first measurement of the ^{235}U and ^{239}Pu spectra based on the combination of experiments at low- and highly enriched uranium reactors.