I believe you can, but not when it’s off. It may depend on your sleep settings (“wake for network activity” or “power nap”). I’m not an expert though — just my recollection.
Then, would the computer not need to listen rather than broadcast?
PS: I know quite a lot about Bluetooth, although mostly about BLE. I don't see a reason why the device should be broadcasting anything. If it is scanning for BLE devices it can do "passive scanning". Is this about ordinary Bluetooth and is that different?
PS 2: Why is this different for Wifi? When a device is not connected to Wifi it sends probe requests for access points. My intuition would blame Wifi rather than Bluetooth. However, perhaps I don't know enough about old-fashioned Bluetooth. Or the device manufacturers have implementations that do not make sense from a privacy/security perspective.