Desalination of seawater to green the desert

What if solar powered distilleries sat on the coasts of the African Sahara desert and the Chilean Atacama desert? What if these distilleries produced fresh water from brine, and then used solar power to irrigate the desert? For virtually no energy input costs, the greening of the desert could be achieved by solar powered irrigation and desalination. The deserts of the world could then become large carbon sinks, accumulating biomass and sucking up carbon dioxide. This could then render the deserts habitable, and, while some desert could be preserved, vast hectares of new grassland and young forest could serve as wildlife sanctuaries.