F-Droid is an open source app repository, meant for downloading and updating other open source apps.
F-Droid's app also has the ability to share and retrieve apps via local networks, not just across the internet. Instructions linked below:
Quoting instructions from the page;
- Open F-Droid. Tap 'Nearby' from the menu at the bottom of your screen.
- Tap 'Find people near me' to search for compatible devices.
- Once people nearby are found, both you and the contact must select each other.
- Once connected, choose the apps you wish to share with your contact.
- Tap the '—>' button.
- One person will be prompted to confirm the request to connect.
- Tap the 'Install' button next to the apps you want to install.
- Tap 'Install' once more. Then, follow the Android prompts to complete the installation process.
That's the procedure for sharing APK:s locally and installing them. (There are two more steps related to management of installed apps, which isn't related to the question.)
The person who has the most recent version can use this to share the updated apk. Unfortunately it does not automatically compare version numbers. You can however see and compare this manually to select what apps to share.