How much public funding do private universities in the United States receive, directly or indirectly? I am trying to assess how much public funding private universities in the United States receive.

I found a study that partially answers the question:

Taxpayer Subsidies for Most Colleges and Universities Average Between $8,000 to More than $100,000 for Each Bachelor’s Degree, New Study Finds
 (published in 2011):


  Washington, D.C. – Taxpayer subsidies that cover the operating costs
  of most colleges and universities ranges from around $8,000 to more
  than $100,000 for each bachelor’s degree awarded, with most public
  institutions averaging more than $60,000 per degree, according to an
  analysis by the American Institutes for Research (AIR) and Nexus
  Research and Policy Center (Nexus).
  
  Among elite private universities, like Harvard and Yale, the average
  taxpayer subsidy is $13,000 per student per year, while the annual
  subsidy at the most selective public universities, like the University
  of North Carolina, Chapel Hill, and the University of California, Los
  Angeles is more than $23,000 per student annually.
  
  The financial figures reflect the amount of money colleges and
  universities receive in direct government support and tax breaks. They
  do not include loans and grants provided by state and federal
  governments to help students meet tuition costs.


However, this leaves out how much public funding go into loans and grants provided by state and federal governments to help students meet tuition costs in the United States.

I got interested in that question reading U.S. to Forgive at Least $108 Billion in Student Debt in Coming Years (published on November 30, 2016).
Estimating indirect subsidies is a much more subtle question than it sounds like.  There's no way to answer it without some economic modeling, and even if you do that, you'll get a debatable answer.

For example, consider tax deductions for charitable donations.  To what extent should they be counted as a subsidy?  (For scale, current-use giving and endowment distributions amounted to 40% of Harvard's budget last year, so how you account for this really matters.)

Imagine a well-off donor, with a marginal tax rate of 33%.  If they donate $1000 and deduct this money from their taxable income, then the government loses $330 in tax revenue.  Should we count the $330 as a government contribution?

One argument says yes.  Imagine that the donor has chosen $1000 because that's how much they feel they can spare.  If they had been charged the $330 tax, then they would have been able to spare only $670 if they wanted to end up keeping the same amount.  This means the tax-deductible donation is really equivalent to a $670 donation from the donor plus a $330 subsidy from the government.  Under this theory, a substantial fraction of charitable giving is really a government subsidy in disguise.

Another argument says no.  Many donors, especially small-to-medium donors, don't make carefully optimized financial decisions based on the tax consequences.  They probably chose $1000 because it sounded like a reasonable number, and would have made the same donation regardless of the tax situation.  In that case, the $330 in reduced taxes ends up as a reward to the donor for being charitable, and it has nothing to do with the university's finances.  To the extent a change in tax policy wouldn't change the donation rate, it's unfair to count the tax deduction as a subsidy.

Of course these aren't the only arguments one could make on this topic, but let's focus on these two.  Now the question is which argument better describes charitable giving practices.  The truth is undoubtedly somewhere in between: donors are in fact influenced by tax deductions, but don't make all their decisions on this basis.  To give a principled number for how great the effective subsidy is, you'll have to model this effect, and the answer may even vary between universities based on differences in their donor pools.

What makes this whole issue even more subject to debate is that politics comes into the picture.  If you view the $330 as tax money that by default belongs to the government, then the whole $330 should obviously be counted as a subsidy.  If you believe the government has no business taxing charities or donations to them, then none of it is a subsidy because the government never had any right to this money in the first place.  Many people lie somewhere between these views, but some pile up at the extremes.

The net effect is that if you really want to figure out how much of an indirect governmental subsidy private universities get in the U.S., you'll first have to articulate your political position, and then do some economic modeling if politics hasn't settled everything yet.  Whatever you decide, someone will tell you you're doing it all wrong.  There is no answer all reasonable people must agree with.