While Molinari and Proudhon were certainly aware of each other (both published in the Journal des Économistes, for example), neither of them, to my knowledge, ever acknowledged being influenced by the other. Nor is that surprising; they belonged to rival libertarian traditions that had grown increasingly hostile to each other, and the debate between Proudhon and Molinari’s mentor Bastiat in 1849-50, while beginning in a cautiously friendly fashion, had ended in a flurry of insult and acrimony. Nevertheless, I think there is good reason to suspect mutual influence.
While Proudhon declared his opposition to the state as early as in his 1840 work What is Property?, he at that time offered little in the way of detail as to what sorts of social or economic mechanisms might replace state functions.
Then, in 1849, Molinari published both his article “The Production of Security” (in the Journal des Économistes) and his book Soirées on the Rue Saint-Lazare, in both of which he described how private enterprise could take over the functions of the state. (Molinari’s ideas were subsequently discussed in the Journal; see here and here.)
Two years later, in 1851, Proudhon published his General Idea of the Revolution in the 19th Century, in which he now discussed, in more detail than ever before, his ideas (in part similar, in part different) as to how private enterprise could take over the functions of the state. This is where I suspect Molinari’s influence on Proudhon.
In that same work, Proudhon described the process of transition as the “dissolution of government in the economic organism.” Over three decades later, in his Political Evolution and the Revolution (1884), Molinari used almost the same phrase to describe the transition to a stateless society – the “diffusion of the state within society” – although he somewhat disingenuously suggested that this approach was in contrast to anarchism: “The future thus belongs neither to the absorption of society by the state, as the communists and collectivists suppose, nor to the suppression of the state, as the anarchists and nihilists dream, but to the diffusion of the state within society.” (pp. 393-94; cf. p. 482, where he speaks of the “diffusion of government within society” rather than of the state.)* This is where I suspect Proudhon’s influence on Molinari.
[* Cf. Benjamin Tucker in 1890: “Anarchists work for the abolition of the State, but by this they mean not its overthrow, but, as Proudhon put it, its dissolution in the economic organism.” Tucker was certainly aware of some of Molinari’s later work – see here and here – but whether he had read the particular works by Molinari here named is unclear. Certainly Tucker’s idea for competing security agencies resembles Molinari’s, but Tucker had also read, e.g., Spooner’s similar proposal from 1851, and I doubt that Spooner would have read Molinari.]