If the private company can't build a better network and provide better service than the people doing it for themselves then the private company doesn't get any business.
Even if the private company can build a better network at the same price, the customers of the private network will not just be paying that price--they don't become exempt from taxes, so they'd have to pay double. This prevents private networks from becoming successful.
What you would be saying is only correct if the people can choose to purchase the private network's service instead of the government network's service. This is not an option.