Upvote:2
Short answer: no, it is not legal for an US citizen to work in France, Italy and Spain during a visa free stay.
For France it is explicitly stated that US citizens need a work permit.
Long answer:
Almost all work based rules are desided on a country by country base
Unfortunately, the official EU summary is an unwieldy excel table for which the wiki link below has a summary of.
Assume that this may not be up-to-date, check the countries counsulate sites in your country to make sure that nothing has changed.
https://en.m.wikipedia.org/wiki/Visa_policy_of_the_Schengen_Area
look for:
the summary above is based on this table.