The French Justice Department appoints notaires to their position but the specific notaire who supervises a particular transaction is generally, but not always, chosen by the vendor.
If for some reason you are not happy with the notaire the vendor has selected; perhaps you would like to use a notaire who speaks English for example, then you are quite at liberty to ask to use another one. The decision needs to be agreed by both sides and once appointed the notaire represents both sides equally in the transaction.
Many English buyers also appoint a solicitor either locally or in the UK. Whilst it is not necessary to do this to complete the transaction safely, it might be a good idea if you struggle with the implications of the French process or need advice on the ramifications of the French legal system on your tax or inheritance planning for instance.
The fee payable for the notaire is set out in a national scale of charges based primarily on the value of the transaction. These charges vary between 6% and 8% but are reduced on new build and properties less than 5 years old to between 3% and 4%. The responsibility for paying for the notaire is the buyers and come on top of the price you have agreed to pay, so be prepared for this when you budget on your purchase.