So now that you can access Game Center on your (compatible) iOS device running 4.1 (or higher). Testing your own GC apps is now a bit of a snafu.
Here's what I have found: First, you'll need to get your app setup in iTunes Connect (see Apple Docs). You'll need to test a debug or ad-hoc build, I have found both of these builds will work (in case you see it posted elsewhere that you must use a debug only build).
Once you install your test app on your device;
1. Launch Game Center and tap your email, select "Sign Out" and now close Game Center.
2. Open Settings, tap Store and tap logout (this helped me once and I still do it, not sure if it's required but it certainly won't hurt).
3. Now launch your test app and sign-in with a sandbox test account OR (better still) create a new account.
Note: I use a bogus email account since I don't have email addresses hanging around for testing purposes. I will try to grab and post a screen-shot because once you launch the sandbox version of GC you'll notice the new account setup screen is noticeably different.
First Clue: Login to sandbox: