Brad Plumer's rundown of the promise and peril of infrastructure privatization is a must-read, but in terms of first-best optimal policy solutions it seems to me that this conversation begins and ends with the basic point that it's not in the public interest for access to infrastructure projects to be priced at a profit-maximizing level.
Imagine a city that owns, among other things, a bridge. Sometimes this bridge is very crowded. The smart thing for the city to do is to put a toll on the bridge and set the toll high enough to leave traffic uncongested. One benefit of that policy is that it drastically reduces the quantity of economic waste generated by people sitting in traffic. Another benefit of that policy is that the toll revenue allows the city to decrease taxes, which reduces the quantity of deadweight loss created by those taxes. So far so good. And in practical political terms, a privately owned bridge may well do a better job of remembering to set the price high at peak demand times than would a public agency. But suppose the bridge isn't crowded at nights or on weekends? Well here the optimal thing to do is to make the bridge toll-free. After all, the city has the damn thing and so people may as well use it. Among other things, making the bridge free at points when demand is low facilitates time-switching of trips so that people who want to avoid the toll during peak hours have every incentive to use the bridge some other time. A private bridge owner, by contrast, doesn't want to ever let the bridge be toll-free. He's not really interested in managing congestion efficiently or maximizing the overall social value of the bridge. If he's smart, he'll make the bridge cheaper at low-demand times than at high-demand ones but it's never profit maximizing to offer a free bridge.
Now in the real world, publicly owned transportation infrastructure is essentially never priced optimally as I outlined above. So it's very possible that in practice some jurisdictions are reaping some gains by moving to privatization. But in a structural sense, privatizing your bridges is a step toward locking bad pricing policies in permanently.