Interesting question, and one that opens up numerous areas of history, communications, community, control, freedom, and public vs. private operation.
In the United States, the postal service from its beginning was publicly operated, and served specifically to distribute timely information and general knowledge in the form of discounted rates for newspapers, magazines, and books, as well as providing personal correspondence and parcel delivery. It also established principles of privacy to correspondents.
This was in contrast to various earlier systems, either privately operated or, frequently, operated as part of royal intelligence services.
With the emergence of telephony and broadcast technologies, there were numerous national debates throughout the worrld over public vs. private operation, with many countries opting for primarily public systems. The U.S. chose instead, generally, a regulated private approach.
There are a few histories of these developments, though fewer than you might think. University of Illinois media scholar Roberrt W. McChesney has several books on the topic.
Several countries leveraged their national telecoms systems to create early computer networks, most notably France's Minitel. In the US, the early DARPANET, ARPANET, and Internet were largely public operations, though through a mixed-control operator network, including the Department of Defence, RAND, SRI, the National Science Foundation, and numerous universities, themselves a mix of public and private.
The Internet itself gradually transitioned from research to commercial use between 1985 and 1995, in a not particularly structured manner. See for example:
https://groups.csail.mit.edu/mac/classes/6.805/student-paper...
There remain elements of government involvement and access to both Net infrastructure and firms, including investments in several significant firms (notably Google and Facebook), as well as intercept capabilities through network operators, engineered device and protocol weaknesses, CA compromises, and both law-based and extralegal means, fairly trivially confirmed through a few moments research. (EFF and The Intercept document much of this.)
At the same time, there are countries which do substantively operate or participate in social networks either directly or through public-private enterprises, notably Russia and China.
And much of the objective of networking the world has been accomplished through largely private operators.
In short: they could, some do, there's a great deal of history, and the present regime arose likely in significant part as national governments and agencies thought they were seeing their needs met through the cooptable efforts of others.