The multimessenger observation of gamma-ray burst (GRB) 170817A from the nearby binary neutron star merger GW170817 demonstrated that low-energy gamma-ray emission can be observed at relatively large angles from GRB jet axes. If such structured emission is typical, then the currently known sample of short GRBs with no distance measurements may contain multiple nearby off-axis events whose delayed afterglows could have gone undetected. These nearby neutron star mergers may produce telltale radio flares peaking years after the prompt GRB emission that could still be observable. Here, we show that several short GRBs observed by the Burst Alert Telescope on the Neil Gehrels Swift satellite, with no identified afterglow and no distance measurement, could potentially be associated with radio flares detectable by sensitive cm-wavelength radio facilities such as the Karl G. Jansky Very Large Array. We also examine optical follow-up observations that have been carried out for these events, and find that a nearby GW170817-like kilonova is ruled out for only a third of them.