I doubt if reading all member emails by admin is a breach of privacy because in general privacy laws are concerned with the disclosure of personal information to third parties by the person responsible for collecting and maintaining the information, in this case the admin.This all depends on the particular laws of the country where the breach is alleged to have occured.
When a member joins a site they agree to provide personal information to the site and the site is bound by laws to protect that information, eg not sell email addresses to anyone. It gets very complex when deciding which jurisdiction is to apply, if for example the admin is in one country, the server another and the member another and a breach occurs across border. To this end a clause in the Terms of Service can assist by stating any dispute will be decided by the law of a certain territory.
When the member also joins they agree to abide by the rules of the site and those rules usually provide that infringing activity can will be sanctioned therefore the member is impliedly agreeing that their activities will be monitored.
Whether such monitoring should extend to reading of each and every email I don't think is neccessary and would be better for a member to report to admin any infringing activity whereupon the admin must then act.The act of reading every email seems unneccessary, over-the-top and to the point of snooping and certainly doesn't provide confidence and trust in the site. Members prefer to be in control and should be encouraged to report unsavoury activites.
Again this depends on the terms of service agreement and the actual service provided by the website. For example, copyright cases have found ISP's vicariously liable for the infringment of users because they have knowingly provided a conduit, or in some cases the actual tool, for the infringing activity.
So phdphdphd if your site is promoting illegal activites or shady activities then it is probably in your interest to monitor those activites and communications between members though I doubt that may be enough to save your butt.
If your site is all above board then why waste your time? Let your members do the policing and you make the final decision.