No, it's a government agency funded by the county. The deputies are actually county employees.
A private law enforcement or security firm would operate under a contract with the city, county, or other entity, and the employees would actually be employed by that private company. If they are found liable for damages due to negligence or willful misbehavior, the company or the individual employees would be held liable, not the city or county who contracted with them.
In a true free market anarchy, the company might actually be hired by a consortium of insurance agencies to protect the insured (and the insurance companies) from loss, much the same way a baron would hire individuals to protect his barony back in feudal times.
The concept of a public law enforcement agency is actually a relatively modern concept, begun in London in the 1840's. For most of his law enforcement career, Wyatt Earp operated under a contract, being paid a couple dollars for each drunken cowboy he arrested.
