In this paper, we present a sample of ten short gamma-ray bursts (GRBs) with a robust redshift determination, discovered by the Swift satellite up to 2011 January. We measure their X-ray absorbing column densities and collect data on the host galaxy offsets. We find evidence for intrinsic absorption and no correlation between the intrinsic absorbing column density and the projected offset of the GRB from its host galaxy centre. We find that the properties in the gamma regime (T90, fluence and 1-s peak photon flux) of short GRBs with ‘bright' and ‘faint' X-ray afterglow likely disfavour different prompt emission mechanisms. The host galaxy offset and GRB duration (T90) do not correlate. Instead, there is a hint of anticorrelation between the effective radius normalized host galaxy offset and T90. Finally, we examine the properties of short GRBs with short-lived and long-lived X-ray afterglows, finding that some short GRBs with short-lived X-ray afterglows have their optical afterglow detected. In light of this, the X-ray afterglow duration does not seem to be a unique indicator of a specific progenitor and/or environment for short GRBs.