Sharing files was introduced in Sync v.2.5. Here you can find the overview of the feature, and this article will cover peculiarities of file sharing on iOS.
All shared files - uploaded and downloaded - appear in the list "Shared links" in Sync menu. Additionally, downloaded files are listed in "Downloads" folder in Sync UI.
To start sharing a file tap on "+" -> Share file, select either Photos app or browse for other apps. Multi selection is not possible though this flow, and files are to be sent one-by-one.
By default the generated link is valid for three days and currently this cannot be changed.
To receive a file from remote peers tap on "+" -> Scan QR code and scan the shared QR code.
If you go to "Downloads" folder and tap on "link" icon at the upper right corner, you'll see history of all file download transfers. Tap on each to see details about each transfer.
Removing file from this "Downloads" folder in Sync UI removes the file from iOS device. Transfer history will still show them though.
In "Shared links" menu you can remove it from Sync UI only, and files will remain on the system.