Middle Earth at the end of the Third Age was largely uninhabited.
The ravages of the Witch King's rule still held effect in Eriador, and the general ravages of orcs had a deleterious effect on the economy generally. Just think of the Dark Ages during the Viking era, only the Vikings never stop raiding; the only thing holding them back in the north and west are the Dunedain. Meanwhile, the Elves and Dwarves keep pretty much to themselves.
Gondor suffered repeatedly from onslaughts by the Kingdoms from the East and South. Rohan was an agrarian society, so the population there may have been the greatest per square mile, but still quite thin by modern standards.
By far the densest population was probably the Shire and Bree, outside of a few fortress cities like Minas Tirith and the rather prosperous region of Esgaroth and Dale, now that Smaug was gone. Hope that helps.
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