Roads and trails have a ubiquity and long history in English legal thought as something more related to a freedom than to a consumer good, the way modern drivers see them. But toll bridges and tunnels were quite commonly built and owned by private individuals, in the same manner as ferries were.
In the US, there were a couple of periods in the 19th century that saw “cross-country” roads built by private enterprise. The turn-pikes of the early 1800s radiated from major cities, allowing farmers in the region easier access to central markets and shipping facilities. Farmers resented the high prices and often poor conditions of these facilities, and the growth of railroad connections soon made the turn-pikes irrelevant.
In the Middle West, another burst of activity 1848-1855 saw a half-dozen “plank roads” built radiating from Chicago (the area I’ve researched most closely). The wooden planks, smooth riding at first, deteriorated rapidly or were stolen and the plank roads were soon supplanted by the growing number of railroads radiating from Chicago.
Thirty years later, other privately financed toll roads were established in the Chicago region, this time surfaced in gravel, some of them following the routes of the earlier plank roads. The toll roads were often in poor repair and farmers despised paying to use them. When the owner of one (today’s Milwaukee Avenue) was shot during an apparent burglary in 1888, a court fight over the fate of the toll roads erupted and a mob burned down the tollgate. Courts eventually declared the toll road had been operating illegally since its 1865 purchase.
As with other 19th century transportation infrastructure, it can be hard to determine which, if any, of the private roads were ever intended to be actual ongoing profitable enterprises. There was so much chicanery involved in securing franchises from the governments involved, stock fraud and overvaluation, and skimming of bondholder money through sweetheart construction contracts that the transportation facility itself can seem an unimportant detail in the scheme.
In the 20th century we saw a new era of truly long-distance roads built as quasi-public enterprises, beginning with the Pennsylvania Turnpike in 1940. For reasons of liability, the nominal owner and operator of all these new toll roads was a state commission or authority, but the bondholders were various private and institutional investors.
America’s Highways 1776-1976 offers some good history on this subject.
Sorry, I don't mean to be pedantic but if a road is privately held then it by definition is no longer a public road. It is now a private road. As mentioned there are plenty of private roads right now, usually operating under the blessing of a state government.
I am not an economic historian but I am an economist and give you a bit of overview of why road may or may not be public. The real problem arises in that if a road is built and connected to "the grid" of other roads, then it becomes very difficult to exclude people from using. So lets say I want a road so I build it, but now everyone uses it. I paid for it but everyone benefits even those who didn't pay for it. This is known as the free-rider problem.
The solution to this of course is to either fund roads with public monies (taxes - which is how most roads in the US are currently funded) or impose a toll (which is what the current private roads are now). As far as the history of this solution, I don't know that it is possible to pen down a date or an exact starting event. I can't quote sources on this just general knowledge (so mods feel free to let me know if this is not an appropriate response), but I believe publicly funded roads were a slow development stemming from the state wishing to spur commerce and the need to quickly move troops. No one was building roads so the government started doing it.
Also keep in mind that prior to the car, roads didn't really need to be anything fancy so many "roads" were just paths that were frequently used and Ill maintained. You can actually find some stories about this in the rather odd but intriguing book "Death in Yellowstone." The early chapters talk about the danger of traveling by means of horse.