Public university salary information is usually available online. What about private universities? Can average salary by field and rank be found at least?
Context: US/North America, mathematics
For the US and mathematics in particular, the American Mathematical Society conducts annual salary surveys, broken down by various categories: public/private, highest degree offered in the department, rank, etc. For example, in 2017-2018, at "large" private PhD-granting mathematics departments, the mean salary for new hire assistant professors was $94,675.
It's far from perfect, but you could try looking at AAUP's Faculty Compensation Survey. This won't allow you to search by field, but it will show you average salaries by rank for each institution.