Does allowing Bluetooth discovery consume more power? Privacy issues and of course turning Bluetooth completely off-- all aside, if I enable "Make device discoverable" will that draw more power than not? As long as Bluetooth is "on", is it always drawing the same power regardless?
I've found no hard evidence for or against it, it is basically down to the Bluetooth stack implementation.

The BT stack needs to weigh power consumption vs availability requirements for the operation it assumes the user wants to perform, so it attempts to turn off both transmit and receive circuits whenever possible.

The ideal case is when two devices are connected, and neither expects any other devices to show up; being "connected" means that the devices have synchronized clocks and agreed on duty cycles. For an idle headset, it is totally acceptable if it listens for a short period once every second, as the user is unlikely to answer the phone any faster; if there is work to be done, the phone will send a packet instructing the headset to keep its receiver on.

Going to connected state from power-on is more difficult, as you have two devices trying to minimize the duty cycles on their transceivers while they still need to find each other (and cannot assume synchronized clocks yet). How devices behave now largely depends on how much power they have available and what their purpose is.

A headset will assume that it is being turned on in reach of a phone that it is paired with, so it will listen continuously if one of the devices in its pairing list speaks up; if so, it will reply with a connection setup packet; after a few seconds, it generally gives up and goes to a low-power mode, assuming that it was turned on inadvertently (e.g. in a handbag).

A car hands-free set has lots of power available, so it can listen and actively probe for paired devices continually.

Being "discoverable" only means that certain broadcast packets are replied to if they are received; otherwise, only unicast packets directed at the device are considered. This setting is technically independent from the power policy, but these are often set together by a middle layer responding to an UI request.