There were special magistrates with vast judiciary duties during Roman times which conducted investigations, they also called witnesses and suspects to court similar to how judges do today. Those magisters were called Praetors and their decisions were already acknowledged as a form of common law (praetorium ius) during the late republic, their power shifted during the empire and they eventually turned into another office entirely.

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Now the praetors weren't judges themselves, think of them as very powerful senators second to only the consuls which acted like the two rulers of the republic. They appointed judges and ratified their decisions if they deemed fit, overseeing the judges rather than taking their time to solve cases. They also issued edicts, many of which were the edicts already issued by ex praetors, having to be issued again to be in effect, those edicts were essentially Roman civil and penal law, and the judges had to take it in consideration. The appointed judges were themselves senators, and they conducted investigations, but not magistrates like the praetors, and they weren't career judges, many were often appointed once and never again, so they likely thought of it as americans do jury duty.

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To answer your question, unless we're talking about a politically charged killing in which the praetor could use his influence to undermine the investigation, killing a senator was a crime against the people, since senators were representatives of the republic and people of Rome. Those were public crimes or crimina publica and the praetors presided over those particular crimes because it was deemed worthy of their attention, and the judges were used as jurors to determine the verdict. Witnesses would be called into question, if there are any, even if they didn't see they might've heard, if the senator had known enemies they would be called in for questioning, and their alibi would be investigated, as well as the alibi of all possible suspects. Suspects were forbidden to flee the city during those investigations and if a suspect refused to show up to court he would be instantly found guilty, although he could appeal for a reschedule beforehand. If the suspect was found guilty he would be sentenced to death, although in some rare cases the convict would be sold into slavery or exiled. Now I tried to keep politics out of it, but if the suspect was a high ranking magistrate such a consul or yes, a praetor, they would have had legal immunity granted through them via imperium, meaning they wouldn't have to go to court until their term as magister ended, but the moment a term ended another one could begin, and if the suspect got elected again and that term also offered imperium that means the suspect which might as well be the murderer or the person who paid the murderer could effectively avoid prosecution. Other crimes, such as faulty contracts were investigated by the parts entirely, and the judge would need only choose who was in the right and admit to his demands.

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Crime in the provinces was a different matter entirely, it was left up to the proconsuls, and I have no clue how they went about it, but they ruled mostly autonomously so I'd imagine they just did what they felt like.

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