APKShare lists all installed apps (system and user) and lets you send the original APK file by e-mail, or share it with Android Beam, Bluetooth, etc.
Root access is not required and no permission is used.
APKShare is available on Google Play.
Signed APK's can also be found on GitHub in the Releases section.
If you use Android Studio, you can import the project directly from GitHub.
Otherwise you can build it from the command line with Gradle.
Clone the repo and type:
./gradlew build
(You may need to chmod +x the gradlew script)
The Gradle script will take care of downloading the necessary libraries and will generate the APK's in app/build/outputs/apk.