REST Web Services can be a great way to share data between iOS and Android mobile apps. I recently answered a question on Quora about this topic. Here’s my answer:
There are several different options here. When you want to support multiple mobile platforms, creating a simple REST api is a good way to go. Soap-based protocols are going to be too complex, slow, and difficult to use from a mobile device.
It sounds like you already have the data being sent to a webserver. From there you will need to store the data in a SQL database, such as MySQL or Postgresql. When you want users to see the other scores, you can implement a simple GET request on your same web service which would query the database, and return back the scores of the other users.
When submitting the score, your app can use this GET operation to read the other scores before submitting it’s own score.
Alternatively, if you need real-time scoring updates, you can use push notifications. Both Apple and Google have push notification capabilities in their architecture, but this will add some complexity and server requirements. If you don’t need real-time updates, I’d suggest each app just read the other scores from the leaderboard as needed.