I have a handful of users for whom this isn't working - but it is working for some. It behaves the same regardless of desktop or web.
FWIW, when it does work, you do in fact get a toast notification that the file is downloading.
I'm going to do what I find I have to do frequently: remove, then re-add the users to the various teams where they've been affected. Thankfully, they are team owners (manager types) and don't actually have shifts in the schedules.
I believe I have identified the scenario where this fails: Users are unable to export time clock data from Shifts. The only user who is able to perform an export successfully is the person who actually created the schedule - other users, despite being designated as Team Owners, are unable to export. They click the export button, and nothing happens, no file is downloaded. I have confirmed this behavior across several tenants and users/schedules. It fails in both the web app and the desktop app.
Edit/Update: Confirmed as a bug w/MS Technical Support who was able to reproduce this issue. The only recourse is to suggest this in "User Voice" which seems like a dead-end, surely never to be looked at again. This bug effectively prevents several of my clients from adopting shifts, since there is no way for the folks in payroll, for example, who depend on this data to pull it themselves on demand - they have to wait/hope for the schedule owner to provide it to them, And in many cases, the person who created the schedule initially, is not the actual front-line manager responsible for the schedules.
@Bob Manjoney Thanks for the update, I've just been through a similar exercise in hair pulling. Unfortunately great ideas the get your hopes up with what turns out to be half baked implementations that let you down seem to be an increasing trend for Microsoft these days.
User voice is far too fragmented a forum to get any traction.