This may be common knowledge, but because I had to figure this out myself, I decided to write this up.
I just noticed that changing the password here in community.fantasyflightgames.com does nothing. You still have to sign in to forums with your old password. Also, forum sign out still doesn't work, because asmodee session hangs in background. I have no idea for what the forum password is used.
To change your password which you use to sign in here, open https://account.asmodee.net and change your password there. After that you should probably change your password (to same?) in forum, to make sure you can sign in, if this cross site functionality is someday changed.
And to sign out securely (1), you have to first sign out through forums, and then open the account.asmodee and sign out there also.
(1) If you sign out normally only from forums and close your browser, and later open it again, you can sign in to forums without giving your username and password. This problem affects at least Firefox (probably also Google Chrome).