You need to make choices in defining your question: What is a nation? What is private property?

Presumably you are using nation as a synonym of nation-state, but that concept is only relevant for a short time in history. A nation could be considered a culture or ethnic group, while a state is generally the political control of a coherent territory. Which one do you want?

Private property also needs to be defined. To which extent to you tolerate interference with the decisions of individuals? Within the nuclear family? Within the extended family? Clan? Tribe? Settlement? Nation? To which extent do you tolerate arbitrage by an outside authority? Do which extent to you tolerate enforcing that arbitrage? By whom and which means? To which extent does property extend? Are eg. slaves property or citizens? Do you allow any notion of common interest? Would you allow a lack of absentee ownership, with only possession? How is property established? How is it enforced?

I'm going to have to ask you to clarify your definitions of "privatized." Because if you're talking about the provision of services through private contractual relationships, you can find examples of nearly everything through history being provided through private contractual relationships.

So, tax collection? The fiscal ancien regime in France was built on the "farmers-general," who received a private monopoly to collect particular taxes or customs. It was also hopelessly corrupt, which was one of the major causes of the French Revolution.

The military? There's no shortage of examples of instances where state militaries have been made up of mercenaries. The Byzantine Varangian Guard, for instance, was made up exclusively of mercenaries, under the theory that outsiders like the Varangians would be loyal to the Emperor, rather than have pre-existing loyalties to one or another political faction.

Transport? The backbone of the American transport system was almost entirely run by private railroad companies before World War II. For example, until 1932, New York City's subway system was run by two competing companies: Interborough Rapid Transit, and the Brooklyn-Manhattan Transit Company.

There's no shortage of examples where public services were at one point or another provided by private parties. What isn't present is the kind of libertarian utopia that you propose. (I'm assuming, for the sake of argument, that you're not talking a Somalia-style power vacuum, which is a situation I'm not sufficiently knowledgeable to talk about.)

Sorry, we don't allow throughout history questions. These tend to produce threads which are collections of trivia, not the in-depth discussions about a particular topic we're looking for. If you have a specific question about a historical event or period or person, please feel free to re-compose your question and submit it again. Alternatively, questions of this type can be directed to more appropriate subreddits, such as /r/history or /r/askhistory.