If the 3rd party screws up and gets sued the taxpayer isn't footing the bill so they have incentive not to screw up. Every time the contract renews you have an opportunity to go with a totally new organization whereas even the best efforts at refreshing a government department wind up as a sort of ship of Theseus. A 3rd party also has a damn good incentive not to treat people badly because they are the customer and they have a contract renewal hanging over their head. A 3rd party also has no incentive to spend big bucks operating MRAPs and other military toys that are not relevant to the job at hand. It probably makes sense to retain investigative work under the government umbrella (avoids some conflicts of interests) but for routine patrol duties, security and traffic detail there is no advantage to having official government police instead of private security, at least from the perspective of the people being policed.
A lot of societies current gripes with policing would be mitigated under the incentive structure that having a 3rd party contractor gets you. I'm sure other problems would pop up but I don't think they would equal or surpass our current problems so it would be a net improvement.