How Large of a solar panel would I need to keep a 380W battery charged?

For use on a Electric scooter?

OK with the Panel attached to the top of the Battery case in the Tucson AZ sun. Full exposure. What would be a nice round size. I looked at the ratings but they are not tellin me much.

You're going to want something that gives your battery less than about 130 W or you'll cook it.
That would be about 150 to160 W panel depending on brand and how it's connected.

You may want to oversize this a bit given that your panel won't be collecting all the sun possible mounted to the back of a scooter. If you're not overly concerned about battery life, try a 200 W panel (gotta warn you though - it ain't gonna be small)

Solar panels should have its conversion rating, and maximum power output on their package.

It depends completely on how much sun the panel is exposed to. However, you may be able to determine the smallest possible panel you could use given a certain sun exposure, but you would need to measure the intensity of light at your location first.