There are a range of possible organisational models for hospitality exchange networks.
It seems to me what is appropriate here is a participatory democratic structure. In democratic organisations there is a necessary separation between the general body of members and the representative executive. The general body, which is where the real power must necessarily lie, can decide to elect a representative executive leader and committee, and delegate to them management responsibilities if it feels that is necessary – as it undoubtedly is here. Crucially – the leader has temporary tenure (e.g. 1 year, 4 years etc.) and is answerable to the general body for their conduct.
Another possibility is like Wikipedia. Anyone can edit nearly all articles of Wikipedia. Some people have shown they can be trusted to be able to handle some more power - they can protect pages and block IP addresses in case of vandalism. The international Wikimedia Foundation has other official members, voted by Wikipedia users, and there are foundations set up in other countries and there will be more foundations set up. Every language version of Wikipedia is basically "governed" by people who come with the right arguments and there are votes on decisions that aren´t clear enough. Basically anyone active to some extent can partake in the votes.
The owner, Jimmy Wales, and the developers still are the highest authority. Wales has made some decisions as a 'benevolent dictator'.
"Hospitality Club model"
Basically there is no official organisation, one person has the last word on all important issues. Many people call this a "dictatorship".
Even in an ideal hospitality exchange network there will be some form of hierarchy. But it can be minimised.