In "The Convict Settlers of Australia" by L.L. Robson, it is mentioned that constables (appointed from among the convicts) were armed as there is discussion of punishment for the loss of a firearm, and in "History of Australian Bushranging" by Charles White, all mentions of police indicate that they were armed. On those occasions where bushrangers stuck up and robbed police, they commonly mention that the police weapons were of poor quality and destroy them rather than steal them. The first bushrangers were escaped convicts, later they were native born.

TL:DR Australian police were armed from the first appointees. Initially they were expected to deal with violent escaped convicts or armed offenders out bush.