The original server operated by the Mastodon gGmbH non-profit
It will be open source, end to end encrypted using Signal’s double ratchet encryption protocol, and he plans to make it easy for fediverse platforms to integrate it. The beta will release later this month.
"secure" is relative. They may not be e2e encrypted, but they are still encrypted via TLS, like any HTTPS traffic. It's the same encryption used for online banking. If you care about your instance admin being able to read your messages, you should use Signal or a Matrix client though.
But remember that only a few years ago, almost nobody used e2e encryption, and it wasn't much of an issue.