Most policing is done by private police and private courts, but those are technically outranked by the Republic forces and the Republic courts where the two conflict. Private police and private courts rule everywhere except the capital (and other directly Republic controlled territories), but the Republic Guard (the land-based army of the Dev Council) still has the rights to enforce laws there in certain situations.

For example the Republic Guard has no business entering Solus, the Tri-point private city, without a permission from the ruler of the territory under normal circumstances, but if there is reasonable suspicion that activities in Solus threaten the sanctity of the trade in the region, or that Tri-point is violating the laws of the market, the Republic Guard would be seen as having the justification to enter the territory and arrest who they need to arrest, have those people tried in front of the Dev Council, and punished accordingly.

The Republic Guard (and its naval branch, the Republic Navy) also protect any and and all friendly or neutral trade passing through the Serene Republic waters from foreign pirates, but it does not interfere in privateering that is conducted between two rival /nasfaqg/ megacorporations or their foreign allies, as long as the piracy isn't indiscriminate and/or out of control.

Or so I have understood it.