Tap the "Edit button" on the twitter source you want to configure. Then, change your social parameters.
You can enable Timeline notification if you want, or disable other types of notifications.

Timeline notifications are disabled by default, as this generate too much notification for many users, but it depends on the number of people you have in your timeline. They can only be enabled if you follow less than 1000 people.

Direct message are not yet support as the token provided by Twitter for iOS defined account does not allow reading Direct Messages. There is a feature request to support Direct Message notification.