It's entirely possible and happens all the time that a fraudulent document is notarized.
This means that when you are done with the signature, all legal aspects of the transactions are covered.
Since what needs to be checked is completely standardized and what is in the contract is standardized as well, you are not expected to negotiate anything (except for the price). There are no surprises in the contract either - it is all standard (and boring).
When you are at the notary the contract is read out loud but for one you do not understand everything, and for two you do not worry either as this is the same thing everytime and cannot be neither less nor more than what is required by law.
BTW, each party can have their own notary (what each of them do is legally defined as well and they share the fee (also according to the law :)) or you can have the same for both.
It is very comforting for a stressful act like this.
This is very similar with labour contracts (no notary involved, but a fixed set of points in the contract, no more an no less)
This is typical in the US, too (just called a lawyer, not a notary) but it's not required
If you're buying a house with a mortgage, you're going to be represented by a lawyer who makes sure everything is on the up and up (and more importantly the mortgage company is being represented to make sure everything is on the up and up)
But, it's not required and really all that's required to transfer property is a signature on a deed
This is actually a two-step process. You sign a first contract that entirely engages both parties - the only exception is if the buyer cannot get a credit. They have a month (or something like that) to secure it.
Then once the funds are there the final contract is signed.
There is a legal possibility to negotiate part of the notary cost (to lower it below the legal maximum) but the chances are thin (they get better for expensive houses).
The notary has a monopoly and charges for it a lot (the amount is regulated as well) but at least you have no legal worries.