For example, if you murdered somebody in Roman Empire could you expect regular soldiers to come arrest you or specialised internal civil order units (police)?

Neither, generally. The Roman world did not know anything like our modern conception of a police force, military or not. This is a surprisingly common question, and maybe one that I should add to the FAQ (although all the answers would be my own :/) but I address this problem here and here, as well as on a number of other posts--but those are convenient. Roman law did not generally provide for lengthy periods of arrest and citizens were not usually held confined before their trials--in the life of Gaius Gracchus, for example, Plutarch (possibly drawing from an actual speech of Gracchus') mentions that even in capital cases the defendant was allowed to remain at his home and was summoned in the morning by a minor official. In the rare cases, such as the Catilinarian conspiracy, when individuals were held awaiting trial (or in the case of the Catilinarians not awaiting trial) they were held frequently by private citizens, who were required to ensure that the defendant would appear in court.