I can't be any more specific, Michael, never having worked in a county prosecutor's office. My view is from the other side, the defense perspective trying to retrospectively piece together police investigations.
I am certain the practices vary from state to state, jurisdiction to jurisdiction. But in New Jersey at least, having reached a certain stage in the case (which I can't specify precisely), the prosecutor's investigators take over most aspects of the investigation, or, stated differently, the management of the case under the prosecutor's direction. This would include, most prominently, final witness interviews, and the taking of custody of physical and documentary evidence, and assuring they're processed correctly. Others items such as follow-up canvassing and the general following of raw leads to other witnesses and evidence MAY fall within the prosecutor's domain, or they may stay with the police, depending on the jurisdiction.
So far, as has been relevant to my cases, I have observed no infighting that would create a problem for law enforcement. Either the lines of demarcation are clearly drawn, or the elans of the different units breed a cooperative attitude.