We present multiwavelength observations of the Swift short \(\gamma\)-ray burst GRB 231117A, localized to an underlying galaxy at redshift \(z = 0.257\) at a small projected offset (\(\sim 2~\)kpc). We uncover long-lived X-ray (Chandra) and radio/millimeter (VLA, MeerKAT, and ALMA) afterglow emission, detected to \(\sim 37~\)days and \(\sim 20~\)days (rest frame), respectively. We measure a wide jet (\(\sim 10.4^\circ\)) and relatively high circumburst density (\(\sim 0.07~{\rm cm}^{-3}\)) compared to the short GRB population. Our data cannot be easily fit with a standard forward shock model, but they are generally well fit with the incorporation of a refreshed forward shock and a reverse shock at \(< 1~\)day. We incorporate GRB 231117A into a larger sample of 132 X-ray detected events, 71 of which were radio-observed (17 cm-band detections), for a systematic study of the distributions of redshifts, jet and afterglow properties, galactocentric offsets, and local environments of events with and without detected radio afterglows. Compared to the entire short GRB population, the majority of radio-detected GRBs are at relatively low redshifts (\(z < 0.6\)) and have high circumburst densities (\(> 10^{-2}~{\rm cm}^{-3}\)), consistent with their smaller (\(< 8~\)kpc) projected galactocentric offsets. We additionally find that 70% of short GRBs with opening angle measurements were radio-detected, indicating the importance of radio afterglows in jet measurements, especially in the cases of wide (\(> 10^\circ\)) jets where observational evidence of collimation may only be detectable at radio wavelengths. Owing to improved observing strategies and the emergence of sensitive radio facilities, the number of radio-detected short GRBs has quadrupled in the past decade.