Both sides of any transaction can have separate notaires, should they want to transact the property in that manner. Or, they can also do it through the same notary. Which is what any notary would prefer since they would not the need to share the fees.
France has one of the longest transactions periods. You are lucky to transact within a period of two-months. Why such a long wait? Too few notaries with too few assistants handling too many transactions.
The contracts themselves are a matter of public record. If you thought your mother was not treated fairly in selling her house, then the signed contract could be made available to you by means of a registered letter to the public notary.
There is very little in France regarding the transacting of property that is efficient. And that starts with the Mandat de vente authorizing an agency to sell your property.
I am presently at odds with a real-estate agent over the words “Lu et approuvé, bon pour mandat”, which agents insist be hand-written just above your signature. That additional notation was made unnecessary in the early 1990s by a court’s decision. But it is amazing how some agencies will insist upon it.
My point: There is a very simple Mandat de vente that is printed by a company called Tissot - and you can download it from here. Of course, it is not free but its simplicity (compared to others that are 3/4 pages long) is a good guaranty that you will avoid any confusion. Note, however, that if you are signing the document with more than one agency, the wording changes. So, be careful of which version you are using.
If you want to sign with ONLY one agency, then there is another wording that makes the mandate “exclusive”. Meaning that if you sign it, your hands are tied to the agency mentioned for the period of validity of the mandate. Which typically is only three months.