The fact is that even if police were to even have been an idea at the time, they weren't needed because in both the Republican (506 BC-27 BC) and Imperial (27 BC-AD 1453) eras, the primary forms of law enforcement were through the city guard, rumourmongers, informers, and the courts of the city and Empire.
If you want an example, look at Jesus' trial, where a combination of rumourmongers (Pharisees) and an informer (Judas Iscariot) brought Christ to Pilate, who acted as judge while using the Jerusalemites as a jury (the court.). The guards escorted Jesus and humiliated Him. Another example would be Saint Paul, where rumourmongers sent him to prison (run by the city guard), where he requested a trial in Rome (to spread the faith in Italy) for a court as a Roman.
Though these are all Biblical (though factual), they provide a picture of Roman law enforcement during the early years of Empire. By its end, in 1453, you were suspect if you were a Latin, Sunni, or spoke with a Turkish accent. Also, if you were a traitor, either you were executed publicly, or you were blinded, castrated, or both, and released to either die of infection, or live as a bitter embarrassment and example to all of what crossing the Romans (however limited they were by 1453) could bring.
Tsar (King) Samuel or Simeon (not sure which one) learned this when Basileios II successfully conquered the First Bulgarian Tsardom (Slavic successor the Khanate of Bulgharia by adopting its endonym while not Turcifying) and his entire army by the hundreds save one out of a group of 100 where blinded entirely, while the remaining soldier was left with on eye to return his hundred to Bulgaria.