No, private construction companies build roads under contract from governments, which paid for them with tax money. It's a simple process to imagine a method for accomplishing the same results without government involvement. Alternatives may operate more or less smoothly, but it is clearly possible.
You are right. They could be, but rarely have. I think the reason is worth thinking about. Suppose two groups could not agree on where to build a road, or had competing road projects? There is a need for coordination. I think this is why we usually decide these things at the community level, and why government is the appropriate body to make such decisions. If we could coordinate our efforts and resolve our conflicts without violence, we would not need government. But that is not the world we live in.
There is one further reason to think about. If roads were only built by voluntary subscription, the wealthy would have roads and the poor would not. Given our commitment to equal opportunity, it makes sense to have a neutral authority like the government arrange for the building of roads so that the roads are available for all, not just some. No concern about free riders. All may use the roads.