When I get a notification, my phone has the standard "message received" sound for e.g. Twitter.
iOS 5 has sound settings for Text, Voicemail, Mail Received, Sent, and Tweet.
However, the Tweet sound seems to only apply to outgoing tweets, not received ones. In Notification Center settings, I can turn on "Sound" for each application, but is it possible to change the sound for app notifications globally, or better yet, per-application?
Not so far. You can do decide whether to turn on the sound or not for particular app (if it enables user to turn off the notification sound). But you cannot change the notification sound globally or per-app (not at least in iOS 5).
This is one of the bigger changes in iOS 5. Under Settings -> Sounds you can change quite a few tones including Twitter, Mail and Sent Mail. It's not everything but the major one's are covered and a huge improvement from the previous versions.
They've also added a considerable amount of new tones, and you can now assign custom tones to all of them which previously didn't work.
If you are referring to the notification beep you hear when you are inside the application itself, that will depend on each application if the developer has setup an option to change it. These aren't notifications as such, but rather built into the application and triggered by events inside the application.
Some applications, for example, What's App, will allow you to change the notification in it's own settings.
The short answer to this therefore is that it depends on the application, some can be globally controlled, and others need to have the ability enabled inside the application.
If you don't like the alert sounds that came with that new app you just downloaded, they're actually pretty easy to change—and you don't even need to jailbreak your phone.
Unfortunately, this only works with non-Apple apps that use their own alert tones. If the app is using a tone that came with the iPhone (like Google Voice's app does), you'll need to be jailbroken to change it. Jailbreaking isn't that hard though. If you don't want to jailbreak, here's how to change an alert tone.
