If there is a profit incentive then you run the risk that the police will decide not to provide services to people who can't pay.
If there was an absolute way to ensure that the same Constututional constraints pplied to them as to the government then it wouldn't be as troubling. I'm not 100% sure (IANAL) but I don't think the courts have found that to b the case. With few execeptions, companies can't be found to have violated people's rights.
However, you are also not addressing my point that it doesn't matter financially if the police are employed by a private company or directly by the government if the government is still paying the private company for its services. The government is still paying for the police out of our tax dollars (with the requisite mark-up for profit) but without the direct control & Constitutional protections.
Unless you make it a true "free-market" transaction where people directly contract out police protection then you end up getting less for more money. It happens with government contractors all the time.
If you actually are suggesting a "free market" individual contract then you will have the situation of the poor having no police protection at all because of a lack of ability to pay. This would be exacerbated if, as some other libertarians have suggested, we make the judiciary "free-market" as well. So much for "equal protection".
Here's the thing - no matter how you set up your system, there will always be a self-interest motive. That never changes, the only thing that changes is the currency you deal in.
I don't advocate a privatized police force that sets their own standards. A privatized force must answer to government - that can't be outsourced. I'm just talking about the mechanism whereby that service is provided. Now, you say government contracting doesn't work, and I agree it's broken. Maybe it's not fixable. I think it's broken mainly because the government is less concerned with cost -as a private firm would be - than it is with following a set of procedures designed to cover asses from bottom to top. Government contractors usually provide poor services because they're too restricted by illogical regulations, and competition is stifled once you're on a list of approved contractors. Again, you may be right, it may not be fixable from that perspective, but again, it's a government problem, not a problem of the free market.
I'm willing to agree their may be no efficient way to get our services from the government. I don't think we've fully and creatively explored the other options, however.