This is most likely a problem on their account, not yours. Tell them to follow these steps to receive notifications.
Step 1: Tap the “Twitter" icon to open the Twitter app, and then tap the “Me" tab.
Step 2: Tap the cog icon and select "Settings" from the pop-up menu.
Step 3: Tap "Notifications" to open the notification options panel, and then use the toggle switches to select the events you want to be notified about. The app's settings will update automatically.
Step 4: Press the "Home" button to return to the iPhone's Home screen, and then tap "Settings." Select "Notification" to bring up the iPhone's own notification settings menu.
Step 5: Tap "Twitter" and ensure that the Notification Center toggle is set to "On," If it is not, the phone will block your Twitter notifications, irrespective of the settings selected in the Twitter app itself.
Hope this helps!