go to settings, then notifications, when you turn that on, any apps that send you notifications should show up on the screen and allow you to turn that specific app's notifications on. i have scorecenter and i just tried it, no problem. i have it turned off to save battery though, and only turn it on if i want to check something.
Once you have turned on notifications in settings > notifications on your iPhone home screen, go to the ESPN app under your team's alert preferences and click the button "sync alerts" to essentially tell the ESPN app that you have enabled notifications. In my opinion this should be done automatically on startup but alerts won't work until you do this...