'ES File Explorer' for Android works great. I tested this yesterday, using my Android tablet (just because the screen is bigger, should work identically for Android phones). I attached a USB thumb drive containing a few music, photo & video files to my home router. Once the Android tablet was connected to my WiFi home network, I was able to use ES File Explorer to explore other devices connected to my home router, including the music, photo & video files on the USB thumb drive connected to my home router, and play those files. If using the model I suggested earlier i.e. 1 main directory per user, with directory sub-trees for e.g. photos, videos etc. on the USB thumb drive, it should be easy for users to know where to upload their files, and for other users to know where to find them.
Sorry, I don't play in the Apple ecosystem, so I don't know what the Apple iOS equivalent of 'ES File Explorer' for Android is. A quick internet search reveals maybe 'Files Pro' or 'Goodreader'? Others with more iOS experience may be able to provide advice here.
Also, some routers have their own proprietary solution. My Asus RT-AC66U WiFi home router has 'AiCloud' functionality built in to the router. You enable it on the router, and there are AiCloud apps available for both Android & iOS devices to download. And boom, you're done. Obviously, if one goes this route, in an area without internet, phone/tablet users would have to download the AiCloud app beforehand, while they still had internet access.