A party wall surveyor's duty is to settle the dispute, and acts as an impartial expert, and is appointed in writing, rather that employed.
The most sensible option is for both the Building and Adjoining Owner to appoint a single expert, called an ‘Agreed Surveyor’. This legal agreement means that both Owners must abide by the Agreed Surveyor's award.
If each Owner appoints separate surveyors - which they must do within 10 days of being requested to do so, or risk a surveyor being assigned to them by the other party -  then these surveyors must appoint a Third Surveyor, who acts as an assistant and referee to resolve disputes between the two original surveyors.
The Owners are allowed to contact the Third Surveyor to discuss matters in relation to the dispute.