There are private police forces, we call them the "mafia", and people pay them for "protection". Sure, they're corrupt, but with no government funded police, whatcha going to do about it?
If the suspect is eventually convicted (of something at least as serious as kidnapping) then that conviction retroactively justifies the arrest.
If the suspect is never convicted then arresting them is little better than kidnapping. If the police at least had a reasonable belief that the person they arrested would be convicted then it falls under the heading of accidental harm rather than negligence or malicious intent, which shields them from retribution, but that doesn't avoid the need to pay restitution to make their victim whole.
In short: Be sure you're arresting the right person, and don't do anything to them in the process which you'd find difficult to set right.
Enforcing laws is the point of government, and a free market requires enforcement of laws. This is why countries with a non-functioning government are an anathema to business, not a paradise.
The "free market" (in a libertarian sense) does not permit the initiation of force against people who have not resorted to force themselves. If that is a significant restriction on your police department's activities then there is something seriously wrong with your police department.
> Worse, the police aren't going to arrest people who pay them.
If they aren't willing to arrest anyone who pays them then no one will bother to pay them. Resolving disputes between two paying customers is part of their job description, and they won't retain customers if they're seen as biased or corrupt.