Security of accounts with multiple users of facebook - Force Password

You should be confused. That's definitely not the correct behavior, if I'm understanding what you say correctly.

First, and easiest, is check that you have the "Auto Login Facebook Users" and "Log out of Joomla Only" settings int he Login/Logout Redirections area set to No. The 2nd one especially, may help resolve some of your issues by forcing the user out of Facebook when they logout of your site. Please note, this is only if they use the logout button from the JFBCLogin module.

Now, if that doesn't change anything (and it likely won't because I think you may have bigger issues), let me go through how it should work:
Logging out of your site
* User logs into your site through Facebook
* If they logout of your site (and you have Logout of Joomla Only set to no), they should be logged out of Facebook as well. Please try this, and go to Facebook.com to see if you're still logged in.
* If they then click the Login With Facebook button, the pop-up should appear asking for credentials

Logging out of Facebook in the background
* User logs into your site through Facebook
* They go to Facebook.com in another tab, and logout
* User should remain logged into your site (as a Joomla user), but they shouldn't be logged in through Facebook. This can be determined by seeing if the user's profile picture shows in the JFBCLogin module (needs to be enabled to show).
* After logging out of Facebook, you may need to refresh your page a few times for the avatar to dissappear, this is until Faceboook successfully clears your cookie - this shouldn't take more than about 3-4 page-loads

Re-logging in
In all cases, if you're using JFBCLogin, and the username/password fields appear, when you click the Login With Facebook button it should not update the Usermap of a different user. If this is happening, that's not right.
However, if you're logged into Joomla and see a "Reconnect" blue button and use it, that will update the user's mapping. The reconnect feature is meant to allow a different Facebook account to be associated with the currently logged in user. You can disable reconnections in the JFBCLogin parameters.

Hopefully this is a little clear, but I can understand where it might not be. If none of the above works, rings a bell, or makes sense, post (or private message me) your URL so we can see if there is indeed an issue with how JFBConnect is working on your site or if there's just a mis-understanding/mis-configuration.

Thanks!

Hey Andy/Andrew (?),
I just responded to your PM. I don't think the URL you sent is correct. We'd love to help you get to the bottom of these issues ASAP, and I think looking at the site config is the best way to go about it. Please update that, and we'll be on it.

Thanks!