No. Persistence of the data does not require continuous power. But the black box has a battery-powered "pinger" that produces sound to make it easier to find. The batteries for that is only required to last for 30 days.
For communication under(salt)water, you're pretty much screwed with every part of the electromagnetic spectrum apart from visible light, and even that won't work in cloudy conditions or if the emitter is occluded. Sound is really the best option.
Because infrared doesn't work very well in water, and crashes in water are the biggest challenge for finding wreckage.
Airliners also carry emergency locator transmitters† (ELTs), which, when activated by a crash, transmit radio signals which can be triangulated via satellite. However, ELTs do not work underwater. No ELT signals from MH370 were heard.
