You can currently use Live Mesh (www.mesh.com) to sync files including database files if you choose. However, be aware that synchronization can result in conflicts if the database is changed in two different locations at the same time. I don't think you want your conflict resolution to happen at the entire-database level...
You can also try the Live Framework CTP (developer.mesh-ctp.com - sign-up required) and change your app's data access logic to store your data as entries in feeds. The feed-based data model is quite a bit different from a relational database, so this would be a non-trivial effort. There is no referential integrity, and instead of foreign keys you have hyperlinks to resources. Custom user data is stored as DataContract-serialized XML. There is query support, but only for certain generic fields, not for arbitrary user data. Synchronization conflicts can still occur at the individual entry level, but a detailed history of all conflicts is maintained and you can display this to the user to let them choose which entry to use to resolve the conflict.
Your app can choose to program directly against the cloud API, or you can install the Live Framework Client and program against an identical local API.
The following blog post details the various options for storing data in Live Framework http://nmackenzie.spaces.live.com/blog/cns!B863FF075995D18A!163.entry
You can check out the documentation for Live Framework http://msdn.microsoft.com/en-us/library/dd156996.aspx
There are a number of other good resources (training videos, screencasts, walkthroughs, hands-on-labs, blog posts, etc.) in the sticky threads on the Live Framework forum http://social.msdn.microsoft.com/Forums/en-US/liveframework/threads/