How can one pull the (private) data of one's own Android app?

adb backup will write an Android-specific archive:

adb backup  -f myAndroidBackup.ab  com.corp.appName

This archive can be converted to tar format using:

dd if=myAndroidBackup.ab bs=4K iflag=skip_bytes skip=24 | openssl zlib -d > myAndroidBackup.tar

Reference:

http://nelenkov.blogspot.ca/2012/06/unpacking-android-backups.html

Search for "Update" at that link.

Alternatively, use Android backup extractor to extract files from the Android backup (.ab) file.

I had the same problem but solved it running following:

$ adb shell
$ run-as {app-package-name}
$ cd /data/data/{app-package-name}
$ chmod 777 {file}
$ cp {file} /mnt/sdcard/

After this you can run

$ adb pull /mnt/sdcard/{file}