How much can a Uni like Harvard, Yale, or MIT be paid compared to a State Uni under "FREE" education. Who decides the fees?

The use of "uni" indicates that you are not American. Are you a paid troll trying to stir things up?  Or just someone with too much time on your hands? No one is suggesting that the government will set tuitions. My state provides tuition subsidies to all students with a decent GPA through a lottery scholarship. Money put into education is considered an investment in most countries- not some kind of welfare. 

I would second the answer which which says "free" college plans are limited to state colleges and universities.  I have not seen any proposals to date which would extend government-funded college tuitiion to private universities.

As a side note, such plans are not really free tuition.  Rather, college tuition would be paid by the taxpayers as a whole.  That is in contrast to the current situation, in which students and/or their parents pay for much of the costs of attending.  The situation is similar to the Staten Island Ferry.  No one has to pay to get on the ferry.  All of the taxpayers pay to keep the ferry running, rather than just those who ride it.

When politicians promote the idea of "free" college tuition, they are referring to public universities and community colleges, not private schools like Harvard. 

In the US, the tuition at a private university is decided by their governing board (often titled "Board of Regents") based on recommendations provided by the university president.

For a public state university, the process is mostly similar, although there will also be guidance provided by the state government.

Very few US universities, either public or private, are free.  Some have higher tuitions, some lower, some offer more financial aid, some less, but virtually all charge tuition.

The tuition varies from a couple of thousand dollars to $60,000 or more, depending upon the school.  In general, public community colleges are the least expensive.