How common are self-funded PhDs in the STEM fields in U.S. universities?
From the Survey of Earned Doctorates, a large-scale survey of students who earn doctoral degrees from U.S. institutions, 


In the life sciences, 9% report "own resources" as primary source of support.
In the physical sciences, 3.8% report "own resources" as primary source of support.
In the social sciences, 25.3% report "own resources" as primary source of support. 
In engineering, 3.8% report "own resources" as primary source of support.


See TABLE 35. Doctorate recipients' primary source of financial support, by broad field of study, sex, citizenship status, ethnicity, and race: 2014.