As I understand it, Rome did not have a police force as we understand it. Does that mean that the local legion enforced the laws? Or was it strictly by application from the aggrieved?
What I mean is, if a body turned up after a tavern brawl over some loaded dice, would anybody "investigate"?
Or if, say, a tradesmen was killed by a robber, would anybody official try to catch the robber, or would it be up to the tradesman's family to track down the offender, accuse him of his crime, and demand restitution?
I'm going to drop a bunch of links on you because I've answered this question repeatedly and I don't especially feel the need to reinvent the wheel in the short time between my exams:
https://www.reddit.com/r/AskHistorians/comments/51boik/the_vigiles_of_rome/
To provide some particular points of reflection and at least touch on a couple of your reservations and remarks, the Republic--indeed, the Roman state for the entirety of its existence--lacked a public prosecutor. In court cases, both criminal and civil, individual citizens brought suit. This could be, say, suit against an accused thief (theft in Roman law was a civil offense) by the robbed, or it could be, say, prosecution de vi or de pecunia repetundis (violence and extortion, both crimes leveled mainly against magistrates and promagistrates), usually brought to court by senators. Nobody "investigated" crimes with the purpose of discovering criminals and prosecuting them. For one thing, forensic science was not even in its infancy yet in antiquity (and many more modern forensic methods would likely have been suspected, had they existed, of tyrannical breach of libertas, that all-important concept, by the Romans). For another, the aforementioned lack of public prosecutors meant that no bureaucratic force existed whose sole purpose was criminal prosecution--indeed, there are really only a handful of criminal offenses in Roman law, and civil courts under the praetors handled most suits.
Regarding the police, it's highly important to realize that prior to around the nineteenth century or so the concept of a body existing to enforce laws was utterly alien. Public forces of order did not exist so much to protect the individual from unjust or illegal action, they existed to preserve the body politic and the city from conflagration, disorder, and civil insurrection. The idea of having someone to enforce laws is a recent one. Laws exist and are there to be followed. In the event that they were not followed, the courts existed. In the event that everything failed, more drastic measures existed, particularly when the preservation of the state was involved (e.g. the SCU).