There are Always lawyers and legal representatives
I don't think there would be public prosecutors in the sense of the common criminal justice systems. What is usually the case is that the Powers that be appoint magistrates (usually given a different name), who are then charged with solving whatever cases present themselves and upholding the Law.
It is perfectly possible that Paladins or Paladin Orders are granted a licence to deal with certain kinds of crimes. But usually such a licence would stipulate that individuals can appeal their verdict in case of questionable circumstances. (i.e. not being caught in the act, or not having a really good justification story)
Trials with Jury members aren't all that common because most of the people in Faerun are illiterate commoners, not to be entrusted with knowing the finer points of the Law. Plus having such a Jury is a check on the powers of the magistrate, and thus by extension upon the powers that be. The Dukes of Baldur's Gate aren't tyrants, but that I wouldn't expect them to use Jury trials except for cases dealing with Nobles. Limiting your power is never a popular course of action. And it isn't necessirarily a good thing either, because it risks the phenomenon of Mob rule, where a mob of people become the justice system, rather than the just law of the land. (because contrary to commoners the nobles are supposed to be wise and intelligent people, which by extension should also hold true for their appointed magistrates)
Additionally Its a class justice system. In most-all mediëval systems, and in Faerun, Aristocrats, Knights and Nobles were entitled to a Jury of their peers. (while commoners aren't) The reason for that is that the powers that be are themselves aristocrats, and would want a trial by peers. And also in no small part because these peers are very powerful and important to the well being and functioning realm. (You want them to feel safe in your realm, so they don't cause trouble)
It is furthermore the case that Institutions like Unversities and Churches have their own codes of Law and are either completely or to a limited extent excluded from the law of secular authorities. (In this case the Dukes) Such an exclusion is however Always granted by the said secular authorities, so they can also decide to undo it or form an exception.
In actual European history during the mediëval period and later, Universities operated under so called student statutes. This meant students were exempted from local laws and had their own courts, run by their own peers. In practice this meant they could do whatever they want so long as they didn't cross the powers that be. Many students actually got away with murdering commoners as a sport. (because you weren't a man until you killed someone)
They were also allowed to carry weapons in the City, like nobility.
Such student statutes were actually a necessity, not a frivolous measure, because many students came from foreign countries. This was a problem because international law at the time, and still today actually, legally recognises the right of revenge. This meant that as a foreigner in country X you could be held accountable for any acts of your country and fellow nationals. In practice this meant that country X would hold you hostage until your familiy paid ransom for crimes committed by someone else from your country. Since Students needed to be from wealthy families to be able to afford to study, they were desirable hostage targets. And that in turn hurts the University, which hurts the City because it means less students will come.
This might sound like a tangent, but in Faerunian terms such a situation might well apply to instutions like the Churches, Shipping Conglomerates, Universities, Schools and Mage Guilds. A city like Baldur's Gate wouldn't like having its commerce, churches, schools or Mages relocate elsewhere. To prevent that it would be prepared to offer concessions. It is likewise with Paladin orders, whom are actually unpaid elite policemen from the perspective of the City.
Hope that helps
