Since the vault is end-to-end encrypted, it shouldn’t matter where it is hosted, even if it is in the cloud. Here is what a security researcher and a password cracker Jeremy M. Gosney has said about this after the LastPass incident.
”Is the cloud the problem? No. The vast majority of issues LastPass has had have nothing to do with the fact that it is a cloud-based solution. Further, consider the fact that the threat model for a cloud-based password management solution should *start* with the vault being compromised. In fact, if password management is done correctly, I should be able to host my vault anywhere, even openly downloadable (open S3 bucket, unauthenticated HTTPS, etc.) without concern. I wouldn’t do that, of course, but the point is the vault should be just that – a vault, not a lockbox.”
There seems to be a specific problem with registering passkeys, but logging in should work fine if you have already registered a passkey for a site. Not sure if this is the case with all password managers, or just Bitwarden.