This happened to me. I tried two brand new sets of AAs (x6) and it still just beeped saying it was out of power.
Straight to the fix — I had missed part of the user manual that says you should only put Lithium Batteries in your Nest Protects. The manual even goes one step further and says you should only put Energizer Ultimate Lithium Batteries in there.
One Amazon order later and i’ve got some Lithium batteries, now it works great. Hopefully this helps you if you’re googling for the same thing and coming up with nothing like I was!