I can answer this for Sweden.

The older County Laws were in the 1350s replaced with a set of laws for the entire Kingdom, Magnus Erikssons landslag, but the original office of lagman (lawsayer) remained, but became more closely tied to the King - what had been an office elected by the things (assemblies) to know the law and pass judgement became more of a royal office akin to a sheriff or bailiff.

The 1500s is a time of change in Sweden, and during the later part of the century, the court system was established, with häradsrätter, in which the landowning peasants elected tolvmän, a permanent jury that tried cases and passed judgement, led by the häradshövding, a royally appointed bailiff. However, it was common for häradshövdings to hold multiple offices and instruct the tolvmän to elect one among themselves (often the most senior of them) to fill his role as chairman of the court.

During the early part of the century, some justice was still meted out by the things (assemblies) and their lagmän (lawspeakers).

Parishes were divided into fjärdingar, and they employed a fjärdingsman, often a peasant doing this duty part-time. The fjärdingsman was a sort of rural policeman, responsible for bringing in accused criminals, maintaining order and breaking up brawls and so on. If more muscle was needed, he could call other fjärdingsmän or request assistance from the häradshövding and his men.

In the cities with city rights, the burghers were responsible for maintaining order and had their own court, the rådhusrätt (city hall court). In some smaller cities, the burghers themselves patrolled the streets, ensuring that no fire was lit during the night to protect the city from fire and ensured that the often locally ordered curfew was respected.

Larger cities often employed one or several city watchmen that would use blunt weapons to break up brawls and disperse crowds and maintain order.

At the same time, there were a canon law that the church enforced and the priests judged over and meted out punishment for crimes in their respective parishes.

So, if you steal something in a rural area, and was spotted by the owner but got away, the owner would report the theft to the fjärdingsman. The fjärdingsman would interrogate you, and if he found what he considered sufficient evidence (say that the thing you stole are found in your house) you will be brought in front of the thing (if early int he century) or the häradsrätt (if late in the century), where the lagman (if the thing) or the tolvmän and häradshövding (if the court) would sentence you - the punishment was often a steep fine. If you could not pay, the fine was often converted to corporal punishment (such as a rod-whipping) or even the death penalty. Prisons did not really exist, and it was considered too expensive to feed someone during a long prison stay, so being locked in only really happened between the arrest and the sentencing, if you were considered at risk of fleeing your punishment. In rural areas, it was not uncommon for there to be no place to keep criminals, and they were turned over to their families until sentencing, as it was much harder to simply up and leave in the countryside during those times.