We present a systematic search for short-duration gamma-ray bursts (GRBs) in the local Universe based on 14 years 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 $lesssim$100 Mpc and four plausible candidates in the range 100 Mpc$lesssim$$D$$lesssim$200 Mpc. Although affected by low statistics, this number is higher than the one expected for chance alignments to random galaxies, and possibly suggests a physical association between these bursts and nearby galaxies. By assuming a local origin, we use these events to constrain the range of properties for X-ray counterparts of neutron star mergers. Optical upper limits place tight constraints on the onset of a blue kilonova, and imply either low masses ($lesssim10^{-3},M_{odot}$) of lanthanide-poor ejecta or unfavorable orientations ($theta_{obs}gtrsim$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% 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 $lesssim$2.0 yr$^{-1}$ (90% confidence interval) to the rate of nearby events detectable with operating gamma-ray observatories, such as Swift and Fermi.
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