1650s, "a lurching or swaying," from swag (v.). Meaning "ornamental festoon" (1794) is said to be probably a separate development from the verb (but see swage). Swag lamp attested from 1966.
Colloquial sense of "promotional material" (from recording companies, etc.) was in use by 2001; swag was English criminal's slang for "quantity of stolen property, loot" from c. 1839. This might be related to earlier senses of "round bag" (c. 1300) and "big, blustering fellow" (1580s), which may represent separate borrowings from the Scandinavian source. "The primary meaning was 'a bulging bag'" [Klein].