Who can be appointed
Anyone who is not a party. The Act sets no formal qualification, so in practice owners appoint people experienced in party wall work, often members of a professional body. See qualifications.
Family and friends
The Act's definition does not rule them out, but appointing one is unwise. A surveyor has to settle matters impartially, and the other side may challenge an appointment that looks partial.
The building owner's architect or builder
They are not parties, so they can be appointed. But a builder whose own work is in question is a poor choice, and an architect who designed the scheme may be seen as too close to it, particularly as agreed surveyor.
What owners can do themselves
Owners can serve their own notices, and reply to notices, without a surveyor. It is settling the dispute, by making the award, that needs appointed surveyors.
Common questions
Can my neighbour's surveyor act for me as well?
Yes, as the agreed surveyor for both owners, if you agree. See should I accept my neighbour's surveyor?
Where this comes from in the Act
Links go to the official text of the Party Wall etc. Act 1996 on legislation.gov.uk.