  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.
