We pay the contractors. Everything the government has come from we the people. Private property is older than government, I get your point, bur in fact government is one of the biggest violators of private property.
Discussion
WE dont pay them. We pay the government and it plans the road, hires them and pays them. Do you want to do all of that your self? A direct democracy where everything has to be done by every single individual and if people don't agree nothing gets done?
By your same logic you are not paying them either. Your boss is paying them. Because they pay you, you pay the government and the government pays the contractors.
Also I always find this contrast hilarious. Most of the people that argue the same way you do get really angry when you get called a Communist. But your argument sound perfectly like the Communist argument where the people own the means of production because they are the ones to operate them