It seems there are three ways of testing the docs:

1) ./python -m doctest Doc/library/json.rst
2) make doctest (a.k.a. sphinx-build -b doctest)
3) http://sphinx.pocoo.org/ext/doctest.html

Manually running 1) or 2) and fixing things seems okay for a first step, and when everything is fixed, we could add automation to prevent regressions. Georg is in the best place to say how to do it (through a thin test_docs.py integration layer between doctest and regrtest, with sphinx.ext.doctest, or by editing the test target of the makefile to run make doctest -C Doc).