ABSTRACT We present a systematic search for short-duration gamma-ray bursts (GRBs) in the local Universe based on 14 yr of observations with the Neil Gehrels Swift Observatory. We cross-correlate the GRB positions with the GLADE catalogue of nearby galaxies, and find no event at a distance ≲100 Mpc and four plausible candidates in the range 100 Mpc ≲ D ≲ 200 Mpc. Although affected by low statistics, this number is higher than the one expected for chance alignments to random galaxies, and possibly suggests a physical association between these bursts and nearby galaxies. By assuming a local origin, we use these events to constrain the range of properties for X-ray counterparts of neutron star mergers. Optical upper limits place tight constraints on the onset of a blue kilonova, and imply either low masses ($\lesssim 10^{-3}\, \mathrm{M}_{⊙ }$) of lanthanide-poor ejecta or unfavorable orientations (θobs ≳ 30 deg). Finally, we derive that the all-sky rate of detectable short GRBs within 200 Mpc is $1.3^{+1.7}_{-0.8}$ yr−1 (68 per cent confidence interval), and discuss the implications for the GRB outflow structure. If these candidates are instead of cosmological origin, we set a upper limit of ≲2.0 yr−1 (90 per cent confidence interval) to the rate of nearby events detectable with operating gamma-ray observatories, such as Swift and Fermi.