  Charged currents are probed in low-energy precision $\beta$-decay experiments
and at high-energy colliders, both of which aim to measure or constrain signals
of beyond-the-Standard-Model physics. In light of future $\beta$-decay and LHC
measurements that will further explore these non-standard interactions, we
investigate what neutrinoless double-$\beta$ decay ($0\nu\beta\beta$)
experiments can tell us if a nonzero signal were to be found. Using a recently
developed effective-field-theory framework, we consider the effects that
interactions with right-handed neutrinos have on $0\nu\beta\beta$ and discuss
the range of neutrino masses that current and future $0\nu\beta\beta$
measurements can probe, assuming neutrinos are Majorana particles. For
non-standard interactions at the level suggested by recently observed hints in
$\beta$ decays, we show that next-generation $0\nu\beta\beta$ experiments can
determine the Dirac or Majorana nature of neutrinos, for sterile neutrino
masses larger than $\mathcal O(10)$ eV.
