Here are couple of tricky situations to consider when agreeing to a joint surveyor. Both happened to us.
Firstly as building owner, we had a joint surveyor with our previous neighbour who wanted to hold up the works to a time that suited him better. We thought that the party wall process had a set timetable so that couldn't be done. But hey presto, when you have a single surveyor there is no way to force them to act! The party wall act has a mechanism for dealing with surveyors who sit on their bottoms and do nothing, not sticking to the timetable for preparing an award. This doesn't work when there is only a sole surveyor appointed. The sole surveyor was our neighbour's choice as the adjoining owner. We went along with that to save costs (silly us). It cost us in the end as we had to delay the builders and wait for the sole surveyor (a friend of the neighbours, we discovered) to issue the award in his own sweet time to suit our neighbour). We couldn't even use the complaints procedure of a professional body as, although he was supposedly a member of the Faculty of Party Wall Surveyors and appeared listed on their website as a member, it turned out he was not and hadn't been for a long time (e.g. a couple of years). We were advised by the faculty to call them to check if someone is a member as they (clearly) don't update their website and members list very often!
Second tricky situation happened when we were the adjoining owner. New neighbours of ours turned on the charm and of course we wanted to be accommodating and save them costs so had an agreed joint surveyor. However, we didn't bargain on the fact that they would a) ignore the surveyors requirements to safeguard our property from damp, structural movement etc and hence breach the party wall award b) that their builders were cowboys with only a mobile phone number, no address and that the neighbours left them to it and didn't check what they were doing and c) that they (with full knowledge) would allow their builders to trespass on delicate parts of our roof causing expensive damage and d) that when the party wall surveyor indicated he was investigating damage at the end of the job they would hire an army of lawyers and experts to dispute it all. Our single surveyor clearly felt this was more than he had bargained for and swiftly backed down on everything leaving us with unexplained damage that wasn't in the schedule of condition and various aspects of the build not complying with the party wall award but with no requirement for rectification. The law is such that the single agreed surveyor acts as judge, jury and everything else with no comeback on a lot of the decisions they make (e.g. if they decide there's no damage when its apparent to see!). The only opportunity you have to dispute anything is if the surveyor makes an 'award' at the end of the job but ours didn't so there was no dispute on his decisions possible and they were bizarre.
The party wall act has the concept of a 'tribunal of surveyors' so that one surveyor cannot act as a rogue or make stupid decisions. There are two others available to ensure balance, professionalism etc. This also adds credibility to decisions when there are big issues at stake such as subsidence, serious damage etc. It keeps the dispute out of the courts so the building owner saves on this expensive cost but the 'tribunal of surveyors' ensures that there isn't one individual with too much power making decisions which cannot be disputed.
So, do watch out for the flaws of using a single joint surveyor. You lose some of the safeguards when you do this. This can be very costly. We are considering a law case for negligence against the agreed surveyor and a complaint to the RICS. That is our only comeback and perceived wisdom is that it is hard to prove negligence. I'm sure if we had had two surveyors and a third surveyor to resolve disputes we would never have been in this position.