"Institution" is a generous term.
I think it makes sense to keep them private, as long as you regulate them, provided regulation costs less than it would for the state, city, or feds to run the "service" (which it should).
In 2002, San Antonio regulated them down to ~1/3 ($85) of what they were originally charging ($250, the max allowed by state law), on top of $40 worth of other fees.
Couple years ago, my car got towed when I parked under a vacant covered parking spot in my own apartment complex. So I show up at the impound lot to ante up, and not only do they have Plexiglas, but an entire system meant to protect the employees' security; Several dozen video cameras, heavily barred pass-through cashier slots, extremely intimidating signage, and a screening system conducted by what were essentially bouncers before you were even allowed to enter the building. I payed my $290*, and realized that the security was justified due to the rage induced by the injustice of this absurd arrangement. Meanwhile, the owner of this fine (bonus pun!) establishment is probably miles away, laughing all the way to the bank.
Anyway, San Antonio has already caved into letting the tow companies double the fees up to $177.
This is another perfect example of b_b's article on Poverty Capitalism.
*$290 is a LOT of money in San Antonio.
Edit: Holy shit! A little more research revealed that the company that charged me $290 did it illegally, and the owner got busted. That is the picture of the impound lot from which I retrieved my car! Wild.