I don't know how they do this, but Paris is full of Americans living there.

I think they stay for a few months. Maybe they just don't tell their company and the company still think they are working in the same place.

French immigration is currently allowing remote workers on the one year visitor visa, so long as your company doesn't have a presence in France. Basically, as long as you are not doing business in France or taking a job from a local they are fine with it.

However, it's difficult to proceed to a residency permit in this situation, and you can't join the national healthcare system.

Visa-wise, they are probably just committing fraud by staying on a tourist visa.

VPN. Simple as that. Most companies aren't bothering to check anyway, most that do aren't detecting VPNs, and for the few that do that, there are ways to circumvent detection if you are really determined.