Hosted Database for Mobile Game

Hello!

I am developing a game for iOS and Android mobile platforms. I am curious about the database side of things as the game I am developing has a multiplayer feature in which a player would play a level and send its score to another player (like in a challenge format). When the other player plays the game, the app will query the database and finds a challenge has been issued to him. He then accepts the challenge, etc.

For this feature should I be using a SQL database hosted by, say, Microsoft/Amazon? Or are there other avenues for this?