It is not that the French don't care, but rather that French public health authorities are understandable with foreign visitors.
Everyone can do so in their own way, but there are regulations in the French islands which states :
1. Upon arrival, pets - dogs, cats - must be presented at the request of customs authority for documentary checks. You have to justify that your pet fulfill the health conditions - health record with up-to-date vaccination and microchip - established by the Regulation n° 576/2013 dated June/12/2013.
2. The person accompanying must be the owner.
Also be aware that some beaches in French islands have “no dogs” signs, even if there is no much control about that.