[dastx]

Not sure why I'm getting down-voted. It doesn't work in Firefox's private mode. Nearly 4 years after the issue was raised. It was completely dismissed as "something Mozilla needs to fix" on multiple occasions.

[ssklash]

It works, but you need to right click the field you want to fill and mouse over to Bitwarden.

[dastx]

That works, but the right click menu only includes logins. In addition, if your vault is locked, you can't unlock it without switching to a normal window.

[Karupan]

Works, but unfortunately the vault has to be unlocked in a non-private window first.