Over the course of many gaming runs it can be tough to remember what your character has and hasn't learned of what you know, particularly in moments of revelation. I've seen plenty of players start exclaiming about two pieces of info they just put together, only to turn completely red when someone points out that they shouldn't know about one of them. Player intuition and guesses can sometimes imitate the use of out of character information. In some of these cases it can be impossible to make the judgment. This is why, unless you know a certain player has a tendency to cheat, it's a good idea to give players the benefit of the doubt and to ask before punishing.
Some games have abilities, skills, advantages, etc.
Tell your players that if they don't buy one of these abilities up, then they won't know anything other than the bare public details of a group, even if they're a member of that group. It makes it easy for you to rule on what their characters should or shouldn't know of the world background. In these cases, you could ask a player to make a simple intelligence check for her character. If the character succeeds, she thought of the connection. If she doesn't, she didn't.