  We report the results of an extensive radio-continuum observing campaign of
host galaxies of short gamma-ray bursts (GRBs). The goal of this survey was to
search for optically obscured star formation, possibly indicative of a
population of young short-GRB progenitors. Our sample comprises the hosts and
host-galaxy candidates of 16 short-GRBs from 2005 to 2015, corresponding to
roughly 1/3 of the presently known ensemble of well-localized short bursts.
Eight GRB fields were observed with ATCA (at 5.5 and 9.0 GHz), and eight fields
with the VLA (mostly at 5.5 GHz). The observations typically achieved a
1-sigma_rms of 5 to 8 microJy. In most cases they were performed years after
the corresponding burst. No new short-GRB host with optically obscured star
formation was found. Only one host galaxy was detected, the one of GRB 100206A
at z=0.407. However, its starburst nature was already known from optical/IR
data. Its measured radio flux can be interpreted as being due to a star
formation rate (SFR) of about 60 M_sol/yr. This is in good agreement with
earlier expectations based on the observed broad-band spectral energy
distribution of this galaxy. The 15 non-detections constrain the SFRs of the
suspected host galaxies and provide upper limits on late-time luminosities of
the associated radio afterglows and predicted kilonova radio flares. The
non-detection of radio emission from GRB explosion sites confirms the
intrinsically low luminosity of short-GRB afterglows and places significant
constraints on the parameter space of magnetar-powered radio flares. Luminous
radio flares from fiducial massive magnetars have not been found.
