The electronic structures of the $\mathrm{Ag}T{\mathrm{O}}_{2}$ delafossites $(T$ $=$ Fe, Co, Ni) have been investigated using photoemission spectroscopy. T $2p$ spectra exhibit satellite structures around 6 eV below main peaks, reflecting the strong on-site $3d$ Coulomb interaction. T $2p$ spectra of $\mathrm{Ag}T{\mathrm{O}}_{2}$ $(T=\mathrm{Fe},$ Co, Ni) are similar to those of $\ensuremath{\alpha}\ensuremath{-}{\mathrm{Fe}}_{2}{\mathrm{O}}_{3},$ CoO, and NiO, respectively, suggesting the strong covalent bonding in $\mathrm{Ag}T{\mathrm{O}}_{2}.$ The valence-band spectrum of ${\mathrm{Ag}}_{0.75}{\mathrm{CoO}}_{2}$ shows negligible spectral weight near ${E}_{F},$ consistent with its semiconducting behavior. The extracted Ag $4d$ and Co $3d$ partial spectral weight (PSW) distributions of ${\mathrm{Ag}}_{0.75}{\mathrm{CoO}}_{2}$ reveal a small overlap between these two states, while the O $2p$ PSW shows a significant overlap with the Co $3d$ PSW. Such trends in the PSWs are consistent with those in the calculated partial densities of states of ${\mathrm{AgCoO}}_{2}.$