Much of it was essentially private. In times of war, the Navy would conscript boat-owners and sailors when they needed ships somewhat like the merchant marines, except they still fought under the shipping firms leadership and not the Navy directly. Several old shipping magnates in the Northeast started out this way such as Cornelius Vanderbilt.
There was some Navy that the government always had for emergencies, but the vast majority of the ships and crews were private. Even in WWI, occasional privateer ships were used for the purposes of scouting submarines.
When reports of the severity of British aid to the Confederates became widespread, people blamed privateers, as they should have in my opinion. So after reconstruction people seriously looked for ways to reform the Navy.