Bitwarden is cloud-based with synchronization to local caches. If you want total control over your data with Bitwarden you will need to run the server/cloud side. I'd caution that running a Bitwarden server is not for everyone, as one could make the security worse than the Bitwarden-company-hosted cloud service.

I run Vaultwarden on my LAN, with no public/Internet facing service, and sync only on my LAN.

If you're looking for something that's offline first go for pass [0], gopass [1], or any keepass-compatible [2][3][4] password manager and sync the database yourself.

[0]: https://www.passwordstore.org/

[1]: https://www.gopass.pw/

[2]: https://keepassxc.org/

[3]: https://www.keepassdx.com/

[4]: https://strongboxsafe.com/