The password managers are: KeepassDX (Far Left), KeepassXC (PC version of local), Proton Pass (Better privacy) and Bitwarden (Far Right). Please note that bitwarden does some data collection. See their privacy policy here and their privacy spy rating here.
You can even access your passwords if your server is offline. You just can’t add new entries !!
If your vault is locked on your phone for example it needs to contact the server. But granted, most of the time it is not needed.