1Password for Firefox needs the ability to write to the local database to store data. If Firefox is configured to always launch in Private Browsing mode, then 1Password doesn't have the ability to write to that local database.
If you'd like to use 1Password in Firefox as well as browse in Private Browsing mode, configuring Firefox to not open in Private Browsing mode automatically, then manually launching a Private Browsing window would be your best bet. You can configure 1Password to be allowed in Private Browsing mode in the extension settings page.
