The key phrase is accidentally obtained, which refers to the intent of the system operator. The system operator may read any and all stored messages, but may show to law enforcement agents only those that were accidentally obtained. The system operator may not go back and review messages with the intention of forwarding them to law enforcement officials. If, for instance, a message accidentally obtained is forwarded to the police and they want to see more messages, they must obtain a search warrant. If one's rights are violated by a system operator, the remedy is to sue that person and attempt to collect damages.
Criminal penalties are also a possibility.
One can seek or negotiate any level of desired privacy with a systems operator. Some BBSs give users nearly total privacy, while others offer little or none. It is not necessarily in system operators' interest to offer no privacy, since in that case the government does not need a warrant to search the machine. It is only when users have privacy that the government is required to have a warrant. Of course on-line users should read carefully any existing or intended contracts with service providers.