So, in summary, binaural beats certainly do not work the way the sellers claim, but there's no reason to think they're any less effective than any other music track you might listen to that effects you in a way you like. If they make you sleepy (like they all do for me), use them to go to sleep. If they relax you or get you amped, use them for that. But don't expect them to be any more effective than regular music. If someone you know claims that they are, put them to the test, and bust the myth.
The effects are strongest while you are listening to the tones because your brainwaves are synchronized and tuned into the frequency range you desire at that time. After you've stopped listening the effects can still linger for a while afterwards. The timescale will vary from person to person and be affected by what you do after you've stopped listening.