We would have to try it to find out, but many if not most of the recent government experiments with privatizing public services--security in Iraq, public utilities and in the case of Texas, many of their prisons--have been abject disasters. Because you replace a public service motive with a profit motive. Private companies take advantage of government contracts because the bureaucracy is slow to identify fraud and abuse. I think something similar would happen if Texas went further in these efforts.
This is a question that is hotly debated among political scientists and political activists. In general, conservatives would say that you would get more efficient government and save money while liberals say you would get more corruption and reduced levels (or quality) of service.
Conservatives argue that competition in the private sector would reduce costs and improve quality. They say that government workers have no incentive to do good, efficient work. By contrast, private companies would have such an incentive -- if they don't do good work, they lose their contract.
Liberals say you would get a system where the companies would just cut corners. The companies would do as little as possible so as to get more profit.
It is very hard to say objectively who is correct.