Consider a village near a lake. It is common for the villagers to walk down to the lake to go fishing. In the early days of the community it's hard to get to the lake because of all the bushes and fallen branches in the way. But over time, the way is cleared and a path forms not through any centrally coordinated effort, but simply as a result of all the individuals walking that way day after day. If one villager decided to take advantage of the now-created path by setting up a gate and charging tolls, he would be violating the collective property right that the villagers together have earned.
Public property can also be the product of gift.
In 19th- century England, it was common for roads to be built privately and then donated to the public for free use. This was done not out of altruism but because the roadbuilders owned land and businesses alongside the site of the new road, and they knew that having a road there would increase the value of their land and attract more customers to their businesses. On the plus side, unrestricted access means you can do as you please there, without asking permission, so long as you don't violate others' rights. On the minus side, the difficulty of policing public space means there may well be more irresponsible behavior there. A society that permits both public and private spaces that has public and private roads competing with each other, for example allows individuals to make the trade-off for themselves.