Suppose you tweet something and someone else replies to your tweet...(similar to email) you will get a notification that someone replied to you. If you are looking for constant updates from all the people you are following on twitter....download "Boxcar" in app store and it will alert you of all tweets from your followers.
OP, you're not wrong, if they don't work for you... Mine are not working either.
However, I restored mine form backup, and in ios4 i had twitter notifications turned off.
My hunch is that somehow those settings carried over and I'm in the process of deleting Twitter and re-installing to see if that fixes it..
Just for reference, the default setting for the Twitter app is for you to receive notifications when someone @mentions you, BUT ONLY if you follow them as well. You can of course change this so it notifies you when Anyone @mentions you but it's not default!