This depends, of course, on the highly controversial notion of getting this implemented everywhere. There are three ways governments typically contract with corporate capital. A fixed price contract is when firms complete a public project and the price if fixed without renegotiations. Risk is high for corporate capital in these scenarios. A cost-plus contract is when firms complete projects for a fixed fee plus any costs incurred by the firm.
Risk is lower and the incentive to keep costs down is null.
The happy medium is an incentive contract. Maybe the above bourgeois economic solutions do not seem satisfactory. I think in Phoenix, AZ there was discontent with public trash removal so the gov't allowed the public works to compete with private companies and the public works ended up beating out the private sector in efficiency. This is trash removal, other aspects where public vs. private sectors may be qualitatively different is with respect to healthcare. For practical purposes the extent of competition over public projects between public agencies and private firms depends on the function that each would be serving.