I have been out of the trade since 99. MP were considered specially appointed personnel and had peace officer status on military property, or where there was a military nexus or connection. This meant that MP as long as there was this nexus they had authority, world wide, I conducted investigations and police duties in Puerto Rico for example. It has changed now, but back in the early days of Yugo, MP were the police officers for the UN. CIVPOL, such as RCMP members had no authority and were in effect JAFOs (just another f***n observer), this as I have said has since changed.
Off DND property we did not deal as a rule with civilians unless there was an agreement with a civilian authority. In my day, Suffield had an agreement with the Alberta Gov't to police on the stretch of provincial highway that bisected the base. If we came across an impaired driver off base we would hold for the local police to take over. However, there were JFOs (joint force operations) such as anti-drug teams, anti-organized crime etc that were in effect with combined forces federal/provincial/municipal across the country and MP were regular participants. If you strayed onto our property, you were ours. The only other police force that had equal jurisdiction on our turf was the RCMP, however local police were routinely given permission to come onto the property and conduct their investigations.
Some crimes were always jointly investigated with local police such as murder cases. There was changes in the wind during the 90s to blur borders and jurisdictions following some cases and the public expectations and well optics. I am sure that things have changed greatly since my leaving that Trade. Any current MP personnel please feel free to correct as needed to make it current.