Handing work to a Mac
From the web or the phone, Run on my Mac sends the message to the machine that has your checkout. It picks up the request, runs the discussion with its own seats and its own workspace, and reports back. A discussion you continue this way runs in the workspace it was started in, not wherever the Mac happens to be pointed.
If the Mac is busy, the request waits its turn rather than being dropped.
Watching it
While a Mac is running a discussion, the web and the phone mirror it live: each seat's words as they are written, the commands as they run, the diffs as files change. You do not have to wait for it to finish to know how it is going.
Answering from there
An approval the run needs — a command outside the rules, a write outside the workspace — reaches whichever device you are on. A run started before you left the house does not sit blocked until you get back.
Devices
The Macs on your account are listed in the app. Each one names itself, and every discussion says which machine last ran it.