As of January 1, 2024, the Direction Générale de la Concurrence, de la Consommation et de la Répression des Fraudes (DGCCRF) is responsible for monitoring cosmetics products and establishments on its own. This control was previously carried out jointly with the Agence nationale de sécurité du médicament et des produits de santé (ANSM).
The Agence nationale de sécurité sanitaire de l'alimentation, de l'environnement et du travail (Anses) has taken over the cosmetovigilance and risk assessment missions previously carried out by the ANSM. Since January 1, Anses has been responsible for cosmetovigilance and tatouvigilance. These two systems are based on the declaration of undesirable effects caused by the use of cosmetic products and tattoos. These declarations can be made by healthcare professionals, manufacturers and distributors. Other professionals, such as hairdressers, beauticians and tattoo artists, can also report such incidents, as can private individuals. Preventive measures can be taken to protect consumers.
In a table, the French administration summarizes the changes concerning the steps that players in the cosmetics sector must take since January 1, 2024:

Table taken from the https://www.economie.gouv.fr.