No one at Wire or anyone else has access to your messages.
All messages, media, and calls are end-to-end encrypted. The encryption works transparently in the background and doesn’t need to be activated first.
See Wire’s Security Whitepaper and Privacy Whitepaper for more details.
Note
Using the Wire Drive feature: Messages and calls are always end-to-end encrypted, whereas files and folders are secured in transit and encrypted at rest.