Staff do not browse conversations, but messages are not private from Localsville either. There is no bulk transcript reader and nobody reads your conversations for interest.
Two things do happen automatically. If you report a message, that message is shown to a moderator so they can act on it. Separately, messages are automatically checked for signs of payments being moved off Localsville, and a match can raise the message for moderator review without anyone reporting it.
Messages are not end-to-end encrypted, and Localsville does not claim they are. The Privacy Policy sets out how message content may be handled.
Writing from inside the app is usually quickest, because your request arrives with the context needed to sort it out. Everything is on the Support page.