A private server is hosted in a data-center with complete privacy, and with no user metadata activity shared and including protective monitoring by webmaster Bryan Eagle. The archives will be utilizing a private search engine, purposely not Google-based to ensure private discussion. Private server setup is more expensive, $899 to setup with normal support rates thereafter*.
If you don't mind Google having access to your activity and searches, which allows their AI to track users easily, then this is the option for you. This is the least expensive option to start a group, setup fee is $149.
*Group management fees are paid as a per-user low price subscription per list.