Permissions on the phone filesystem are on a per application basis usually. Sure you can mess with them to give access to multiple applications but this is not the supported scenario. On the other hand, the filesystem mounted at /sdcard is open to everyone and apps can share the same folder/file with ease. If you only want to push/pull from the local storage, use the adb 'pull' and 'push' commands to get the files from the phone onto your computer and then mess with them locally. ADB is part of the Android SDK tools.
See: http://developer.android.com/guide/topics/security/security.html#userid
Still, I'm not sure you could really get the browser generated sqlite database easily if at all. Good luck!