Unfortunately, Apple only can do so much about notifications from other apps [and their tri-tone sound of the notification ]đ, most of the apps themselves have the option as soon as you first open it to set a few different options. usually it will ask you if you want to receive 'push' notifications. if you say yes, depending on the app, it sets three different kinds of notifications: badges, alerts, and sounds.
badges are the numbered icons showing how much of a certain notification [as in how much new mail, for example] is waiting for you in an app. alerts are text pop ups that explain when something happens in an app, and sounds are, in most cases, set to that tri-tone ringtone and happen inside an app when something happens [or finishes loading, its completely dependent on the app].
you might be hearing the sound and not seeing any visual notification from the app if the notifcation setting in that certain app is set to only 'sounds' and not 'alerts', so if you see any with just 'sounds' activated try activating the 'alerts' tab on it too.
good luck!
rizz