How can I tell if someone opened a file I shared from Delta Drive?

Two places tell you. Recent, set to Recent by → Opened, lists what was last opened or downloaded — including by anyone you shared it with — with their address and when.

The Shared screen shows each share's last open and roughly where from, then who did what — for example opened four times, took two copies — or Nobody has opened it yet. Activity on a share opens its full log.

See sharing a file.