How does one prevent a member of staff from leaving?
"I resign."
"You can't. I won't allow it."
"freak."
As far as I'm aware there's nothing in NFL rules preventing personnel staff from taking up new roles in other organisations at any time, and if you think that the teams have any way of preventing knowledge going with those staff then you're even more naive than I already thought you were.
I believe it's called a contract
To the best of my knowledge sports organizations use them, and it prevents other organizations from hiring those individuals.
What would be the point in spending all this money on your scouting department if someone can just walk away anytime they liked and give all your information to another organization?
I'm sure people can move around prior to the draft, but a full blown reshuffle after the superbowl probably isn't the norm