What fraction of STEM PhD graduates end up in jobs that require a PhD? Only a small fraction of PhD graduates in STEM end up as (tenured) professors.  For example, see this question and plenty of other material on the internet.

Many PhD graduates end up wokring in government or industry.  Some of those are research positions where a PhD is valued if not required (perhaps the specialisation is less important), but I also know STEM PhD graduates who work as programmers (quite common), as a miner, in the military, or as a reindeer herder (yes).

Within STEM, what fraction of PhD graduates end up in jobs for which a PhD is a requirement?  And what fraction in jobs for which it is considered valuable, even if not required?

In order not to muddy the waters with graduates doing postdocs still hoping for an academic career before giving up, let's consider the first long-term (5 years+) position or whatever they're doing 10 years after completing the 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.