Perhaps one of the best ways to address this is through what is inferred by two instances in Gargantua and Pantagruel. Inference is the best guide to this answer since this 16th century (1567) moch-heroic classic is parody and satire and fantasy, thus directly states nothing.
The first instance to consider is the peopling of Dipsody by Pantagruel after the Dipsode giants were conquered following their invasion of the land of the Amaurots. Having control of Dipsody, Pantagruel immigrates nearly 10 billion Utopians to Dipsody to populate it.
Utopians hold property in common, as Thomas Moore informs us through the characters of "Moore" (who may or may not represent Moore's own ideas) and Hythloday in Moore's Utopia. Thus if Pantagruel is taken to be a serious character, not a parodied character, the conclusion may be drawn from inference that Rabelais rejects the idea of private property.
But are we to take Pantagruel seriously and are we to embrace the Utopian ideas? Judging from Panatgruel's encounter with the Parisian student in Chapter 2.VI, in which Patagruel acts as the voice of reason while the student is the voice of ludicrousness, it may be inferred that Pantagruel is the thread of reason that runs through the fantasy and parody.
Prut, tut, said Pantagruel, what doth this fool mean to say? I think he is upon the forging of some diabolical tongue, and that enchanter-like he would charm us. To whom one of his men said, Without doubt, sir, this fellow would counterfeit the language of the Parisians, but he doth only flay the Latin,...
Another earlier instance involving Gargantuan when he was a youth studying in Paris is the war that arises as a result of the bakers of Lerne's refusal to sell cakes to the grape harvesters with whom they formerly were happy to trade custom (do business). This war is fought over buying and selling private property. While holding private property is not contested, the right to refuse the sale of private property--or the distribution of the goods of private property--is contested.
How long is it since ... you are become so proud? Indeed formerly you were wont to give us some [cakes] freely, and will you not now let us have any for our money? This is not the part of good neighbours, neither do we serve you thus when you come hither to buy our good corn [grains], whereof you make your cakes and buns.
And while the distribution of goods (buying and selling) from private property is contested, one result of the war is the building of private property in the form of the Abbey of Theleme where private enterprise and the accumulation of private wealth are encouraged.
What can we infer from these two instances? In one case, common property Utopian principles are forwarded through the repopulation of Dipsody with 10 billion Utopians. Bear in mind however that Thomas Moore was never clear in his espousal of the idea of the superiority of common property: "Moore" ends the Utopia narrative by saying their "institutions established ... [seem] quite absurd ... [like] their common life and subsistence with no exchange of money." In the second instance, war occurs over the refusal to freely distribute for a fair price the goods yielded by private property, and private property is the result of the war.
The inferences that emerge are complex ones. Rabelais seems to be suggesting that while Utopian common ownership of property is one solution to the inequities of private property that may succeed among peoples like the Amaurots, another solution may be private property equitably available amongst all, as in the Abbey of Theleme, and equitably distributed amongst all through a fair and equitable commerce with just and fair pricing. Rabelais seems to be suggesting that private property is a social good but that it can be marred and turned against society by greed and inequity, an idea echoed by Moore in Utopia:
it seems to me that wherever there is private property, where everything is measured in terms of money, it is hardly ever possible for the common good to be served with justice and prosperity,.... [(Hythloday speaking) Moore, Utopia]