In the United States, this is a non-issue because US constitutional
law doesn't care about "land", i.e. dry dirt, it cares about territory. The standard interpretation is that "federal lands" are owned by the citizens of the US, but held in trust by the US government. Congress has the authority to sell, buy, lease or give away federal lands. Indian tribal lands are a slightly more complicated matter: that is not federal land, that is the land of the particular tribe, held in trust by the US government.
If a lake dries up and becomes "land", the ownership question is based on ownership of the territory. It might be state land if a lake in a state park dries up; it might be my uncle's land if the lake on my uncle's farm dries up.
The complicated question is, what if the oceans dry up (never mind the obvious problem with that hypothetical): what are the boundaries of each nation? In principle, any nation can declare anything to be "their rightful territory". International law recognizes a number of distances for various purposes, the outermost being about 400 miles (the continental shelf). If Russia claims such new land and defends that claim, Russia owns it; if the US does, likewise.