1. My breakout board for the JDY-08 draws about 9.5mA of power @ 3.3V, and this is approximately true even when it is transmitting.
2. I have a pretty bright LED on the board so that can account for roughly 50% of the power draw.
If I use the worst-case estimate: assuming a smallish 600mAh battery (1.2V) with a 60% efficiency, with another 5mA of power draw from the ATMEGA, that is 15mA from the circuit at 3.3V = 50mW vs 432mWh, this means that a fully charged battery can last about 8 hours of continuous power draw and measurement.
Using an Alkaline battery with 2000mAh would give us approximately a day of measurement. I think this is a pretty good even without any power optimization.