Late to this, but we can say a few words. Some reservations from the start would be that bronze age is a long time, and Mesopotamia, or even wider, Ancient Near East, is too large to make any adequate general observations, and even within old Babylonian law, there are substantial regional differences (Northern v. Southern), that are sometimes briefly referenced due to the difference in Semittic and Sumerian legal tradition.

If we run here with old babylonian period, there was royal administration (the palace and staff), regional and local administration in a hierarchy, but not to mistake this with formal hierarchical or appellate court structure. Vast majority of issues would be addressed through private actions to start a proceeding, certain elements (of what we today say criminal laws, but no such distinction between public/private or criminal/non-criminal that we know today existed at the time) had mandatory reporting. We have cases where people were punished equally as the offender if they did not report the crime (e.g famously The Nippur Homicide Trial*). Courts did have significant powers, even inquisatorial, they could compel witnesses, administer oaths and enforce the verdict, but these agents are just labelled just "soldiers" in the texts we have, an no special investigatory or policing function can be found, not comparable with Egypt, for example, and self-help still played an important part. And not to confuse this, local administrations and officials still exercised their office also in furtherance of public safety, it is somewhat settled that there were public funds for compensation due to public safety deficiencies, among other duties.

For an overview of this, Westbrook´s multivolume A History of Ancient Near Eastern Law is the best venue, and a good comparative on at that.

*Martha T. Roth, Gender and Law: A Case Study from Ancient Mesopotamia in ed. Matthews et al. (1998). Gender and Law in the Hebrew Bible and the Ancient Near East.