An app whose package name is
com.example.app has its data files in
/data/data/com.example.app. This applies to pre–Jelly Bean systems, and to the first user on Jelly Bean systems. For other users on Jelly Bean systems, user 1 has the data files for
/data/user/1/com.example.app, and so on.
However, none of this information helps you to use an external app to sync an app's files. Reading another app's data files can only be done on a rooted phone, which is probably not something you want to do on your child's phone. Even then, you need an app that's designed to work with root access, such as DataSync or AppSync. And even then, it's possible that the Kidlogger app keeps its data in encrypted form, to force you to use their sync service, or as a precaution against tampering with the logs. After all, if the sync app can get access to the logs, there's nothing to stop the user of the phone doing the same thing.
The only reliable and secure solution would be to ask the developers of Kidlogger to add a feature that syncs the app data to Dropbox or an online service of your choice: one the user of the phone can't access.