Assuming that you have a database and a content provider in your application....
You can use the database of other application if it is exported. To export your database you just need to set the value of android:exported attribute of Provider tag to true in android.manifest file
<provider
android:exported="true"
android:name="<content provider>"
android:authorities="<content provider authority>" >
</provider>
In the other application's Android.Manifest file provide the permission to access the content provider.
see the links below for more informationContent Providers
and Using Content Providers