As a general rule yes neighbours will go halves in the cost of a new shared fence. This is governed by the dividing fences act which you can access here https://www.legislation.nsw.gov.au/#/view/act/1991/72/full

The invoice will be sent to the person who arranged the quote. They will be responsible for paying the invoice total in full. If you would prefer to each be invoiced separately you can email to arrange this as long as you have written evidence from the neighbour that they are willing to do this. 

You can access documents from the Law Access website to send to your neighbours so the agreement is in writing. We highly recommend that out of courtesy if nothing else you have these completed. The forms can be accessed  here – https://www.lawaccess.nsw.gov.au/Pages/representing/lawassist_fences/lawassist_cantagree/lawassist_fencingnotices_fences/lawassist_fencingnotice_stepbystepguide.aspx