Privatization isn't about efficiency, it's about private profit and reducing services. Private companies have a profit overhead to cover that the government doesn't. In most areas that difference is too big for just "efficiencies" to bridge. The only way to cover that gap is to provide less.
I will use the privatization of schools as an example. In public schools today, the schools are required to provide free lunches to students in poverty and to provide special services to learning-disabled children. Free lunches are an expense on the schools. Extra teachers and aides and special accomodations are an extra expense on the schools.
Charter and private schools are not required to provide free lunches. They are not required to take all students as public schools are. They do not have those expenses. So if the state pays a public school $1000 per student and a charter school $1000 per student then the charter school will have much more of a profit because they don't provide all of the services that the public school does. For those trying to convince the American public that privatization is a good thing they will say that instead of paying $1000 per student, they will pay the charters $900 per student, saving the taxpayers $100. They don't mention that the expenses at the public school are $1000 and the expenses at the charter are $700 per student so the charter is still making a huge profit.
Meanwhile, as the charter schools cherry-pick the "cheap" kids from the public schools (those students without learning disabilities for example), that leaves the public schools with a greater percent of "expensive" kids. Now the public school's expense is $1100 per student. The lawmakers complain about the public school's budget and point to the charter's $900 cost and change the laws to move more kids to charter (for example, by providing busing services to the charters or just establishing new charter schools). Charter schools are still cherry-picking so once again the public school is left with the more expensive students and the cost goes up again.
It is a death spiral that is meant to destroy public education while making the owners of charters and private schools rich. No one's taxes are going to go down. They never do.