I can fairly confidently predict this will not happen like it did for software. Chemical analysis has been around a long time and remains difficult for experts to do accurately without context, let alone for a layman. Gas chromatography, for example, requires large and expensive machinery and some idea of what the substance is composed of in order to determine the concentration of analytes.
Reagent testing is cheap, simple, and straightforward, but it is generally only capable of detecting whether or not some class of substances are present above a particular concentration. You cannot use reagent testing to determine "how pure" a medicine is, let alone whether the impurities (which there will assuredly be) are potentially harmful.
As is currently the case for illicit drugs, I imagine there will be an ecosystem to verify that A) the active ingredient is actually present and B) some limited range of problem impurities are not present, but that is a much less stringent form of quality control than pharmaceutical companies perform.