True, or course there’s some risk. It could also happen the other way around, sign first buyer pays after— either way the buyer and seller need to trust each other. Otherwise we have the fiat system to fall back on, but then there are more taxes, more fees, and more headaches involved.
We did the same thing when we bought the place- came to an agreement with the previous owner, paid cash directly to them and then went to the notaire to do the official stuff. And on the acte de vente we listed a much lower price and therefore paid almost nothing in fees (less than 500€ total).
So yeah, it’s possible but definitely requires trust in the two parties. If buyer prefers to pay taxes/fees involved in return for greater security via notaire, then payment could also be in fiat. But it will end up costing them more.