In civil disputes, both parties agree to have their case heard by a prominent member of Rome's upper classes. When this judex makes his decision, he has no power to carry it out -- but if his decision is ignored, the guilty party can be brought by force to a magistrate and made to answer for his dishonorable conduct. (Despite the name, the magistrate is also not a judge, but an elected administrator of the Republic with various executive functions at his disposal.)

See: https://www.britannica.com/topic/Roman-legal-procedure

> After the issues were delineated and sureties set, both parties agreed upon a judex, who was neither a lawyer nor a magistrate but a prominent layman, to try the case. The proceedings before the judex were more informal: advocates spoke and gave evidence, and witnesses often appeared. The judex made a decision but had no power to execute it. If the defendant refused to pay the fine or make restitution within a certain period of time, he could be brought by force to the magistrate. Then his property could be seized, or he could be made slave to the plaintiff to work off the debt or property claim.

> Was this only for civil disputes or in general?

ius privatum, which includes all of what we would call civil law but also most crimes against individuals. The ius publicum mostly concerned crimes against the state: counterfeiting, embezzlement, treason, etc.