How to continuously keep two databases in sync
This is a textbook use case for standard MySQL Replication.

You will need to decide if one DB will be the master and the other the slave in which case only the master replicates to the slave. If you want clients to be able to connect to either machine and issue writes and then have writes be propagated to both machines than you will need to look into Multi-Master replication:

http://mysql-mmm.org/

If you will be doing just standard master -> slave than MySQL has native tools as thats the default stack.