We consider alternative modes of two-neutrino and neutrinoless double-beta decays in which one beta electron goes over to a continuous spectrum and the other occupies a vacant bound level of the daughter ion. We calculate the corresponding phase-space factors of the final states, estimate the partial decay rates, and derive the one- and two-electron energy spectra using relativistic many-electron wave functions of atoms provided by the multiconfiguration Dirac-Hartree-Fock package GRAsP2K. While the bound-state neutrinoless double-beta decays are strongly suppressed, their two-neutrino counterparts can be observed in the next-generation double-beta-decay experiments, most notably SuperNEMO.