There doesn't currently seem to be a way to understand which user manually executed a task in QlikView.
The QDS task logs only show something like this:
Triggered by 'ManualStartTrigger'. Id:00000001-0002-0003-0405-0607080a0b0c.
However, the ID is the one of the trigger, not of the user who manually triggered the task.
The Management Service Audit Logs, when enabled, only show who created or modified a task, but not who actually started a task manually.
It would be essential, in high security environments, to know which administrator actually started a task.