What fraction of STEM PhD graduates end up in jobs that require a PhD?
STEM is a big place. 

For  physics, APS reports that 47% of new PhDs go into a Post-doc, 38% take a potentially-permanent position (usually in industry), and only 4% are unemployed. This suggests that at least 86% of physics PhDs are using their PhD. Some of these post-docs won't convert to academic positions, but they are unlikely to end up unemployed.

It's true that some of these industry jobs don't require a PhD, but they are generally preferred, plus they usually count as being worth 5-10 years of experience. So I think it's fair to count the industry jobs as "using" the PhD. 

In other sciences, things are different. In chemistry, ACS reports that 14% of new PhDs are unemployed. On the other hand, 94% report that their job is related to their PhD (so, something does not add up).

In psychology, only 63% are "working full time" (it does not specify on what), but that does not count the 24% who are "working on post-doctorates" (?). Further, 54.4% of the employed PhDs claim that "doctorate in psychology [was] important" to finding their job, with another 24% saying it was helpful. 

See also this  article.

My personal experience as a PhD in industry who is involved in hiring is that there is a huge shortage for US citizens with a STEM PhD and useful skills. By the way, "useful" skills might not mean what you think it means -- even abstract math, statistics, thermodynamics, game theory, etc. can be highly useful if you're willing to apply it to real-world technology.