When pairing a new device (or in this case a new browser), that device doesn't automatically get any keys! When adding a new device, to complete the setup you also have to be able to unlock a password group, for which you need another device that already has keys.
In short, a new device doesn't have the same power as the others from the start, first new keys have to be generated which can only happen if you are able to unlock your passwords.
So just don't setup both the Firefox and Chrome extension and you're golden.