One of the masks in Marie Tussaud's collections was a cast of the famous Count de Lorges, who had been imprisoned in the Bastille for 30 years. When the Bastille was stormed in 1789, he was taken to Mme Tussaud so that she could preserve his face for posterity. She got there just in time, because de Lorges died only a few weeks later. In 1838, when Mme Tussaud wrote her memoirs, she still had de Lorges's mask in her collection.

Unfortunately, no Count de Lorges was ever liberated from the Bastille. He was a fictional character, but the story of his imprisonment was widely believed at the time. The fact that Mme Tussaud had his mask in her collection shows how seriously we should treat the rest of that collection and her stories of how that collection came into existence. Especially considering how implausible it is that she would be allowed to take a cast of executed convicts. The authorities had no reason to have souvenirs made, and especially no reason to do it in secret if they thought the masks would somehow be helpful.

For Marat, the situation is different. He died a hero, and there was plenty of official interest in preserving his memory for the future. At least two masks were made. One was made by the sculptor Bonvallet at the request of the Paris Commune, and another was made by the sculptor Deseine.

Étienne-Jean Delécluze, a student of the painter Jacques-Louis David, wrote that David had a cast made of Marat's face to use for his work. One of the pictures he made based on the mask is similar enough to the copy in the Carnavalet museum, pictured here, that I'm guessing that this is either the bust Delécluze mentioned or identical enough to it that it has to be genuine despite the lack of a source for the bust itself.

Unfortunately, I couldn't find anything about the original molds.

Sources:

Hervé Leuwers and Guillaume Mazeau, « Madame Tussaud et le masque de Robespierre. Exercices d'histoire autour de la médiatique reconstitution d’un visage », Annales historiques de la Révolution française. URL: http://journals.openedition.org/ahrf/13083

Eugène Defrance, 1909, Charlotte Corday et la Mort de Marat, page 218. URL: https://gallica.bnf.fr/ark:/12148/bpt6k58356419.texteImage#

Étienne Jean Delécluze, 1855, Louis David, son école et son temps: souvenirs, page 154. URL: https://books.google.fr/books?id=gHA5AAAAMAAJ&printsec=frontcover&hl=sv#v=onepage&q&f=false