Peer-to-peer file sharing

If the internet is shut down, you can still share photos, videos, documents, some apps, and so forth from device to device, using networks that operate locally (Wifi routers, bluetooth, LAN networks, etc.) Again, having these things set up and tested before the internet is off works best, but needs must.

NitroShare for laptops and desktop machines

NitroShare makes it easy to share files between computers with this software installed that are on a common network. Open source, available for Windows, Mac and Linux, it uses broadcast discovery to find other devices on the local network that have NitroShare installed.

F-droid for Android smalrtphones

The F-Droid app allows you to keep a local Android app store (of apps that allow it) available for sharing through your own Android phone. It's a good way of sharing Briar and other open source software when the internet gets shut off.

Android apps that work on LAN networks

• The LAN Share Tool lets you transfer files to other users on a common local network.
• LAN Messenger - P2P Offline Chat and File Sharinglets users on the same local network share files and have encrypted text chatting. There are also compatible builds for Mac, Linux and Windows computers.
• Lan Chat is a pretty basic texting interface for people on Android phones using the app on a common local network.

Through your smartphone's built in features

The Witness blog's slightly out-dated post on file sharing in an internet shutdown has some good advice on how to just use the features that come standard in your smartphone, and I'll just include some quotes to those points here:

• Share files directly with Bluetooth, Wifi Direct, or NFC: "You don’t need to have an internet connection to connect your phone with another nearby device via Bluetooth, Wifi Direct, or Near Field Communication (NFC) (sometimes called Android Beam on older devices). Bluetooth and Wifi Direct are both wireless technologies that can “pair” two devices without another router or access point in between. WiFi Direct provides a wider range and faster data transfer than Bluetooth, but uses up a lot more power. Meanwhile, NFC has a much shorter range (~4cm) and much slower transfer speeds than either Bluetooth or WiFi Direct, but connects faster and uses less power, so can be useful for small transfers when have both devices in your hands."

You will want to be careful who you share access with at this level, and likely turn these things off when not in use as a safety measure.

• Share files with a wireless drive or via a Wireless Local Area Network (WLAN): "A wireless hard drive or flash drive can be used to share files among a team, or multiple people at one time. The wifi drive will typically come with instructions and/or an app for connecting your phone to the drive, and is relatively easy to use. Remember to set a password on the drive for security."

This one requires someone has one of these around, but if so, you can create local file sharing hubs for trusted users.