Swag on a bead simply refers to a continuous wavy line traveling round the bead. It can be any colour, though I mostly see white or yellow on a dark blue, black, or sometimes ark green. These lines can also be reticella, which is composed of multiple colours twisted together to form a single line.
Certain beads will have two wavy lines that intersect. These are called double swag, which I find even more fantastic than regular swag beads. As an aside, there are blue beads with white double swag in Scotland, and if you block them off at the right point, they create the saltire (the Scottish flag).
I haven’t really seen the swag style outside of Europe, and if it exists, it’s rare enough that I could study Asian/African beads for 5+ years and never encounter the term or the style. As for why it’s not really found outside Europe, I’m not too sure. The design seems simple enough, and these areas were in pretty extensive contact. So it probably has little to do with lack of invention and more with simple preference for beads.
Swag beads in Europe were around for at least 1000 years, if not more. Guido cites them in her book on prehistoric and Roman beads and again in her book on Anglo-Saxon beads. Callmer also cites them in his work on Norse beads. They’re also found all over continental Europe during this period.
Something to remember is that beads can have more than one design at once. You can have zone beads that also have eyes, or swag beads that are also zone beads. Any bead can combine any number of designs, which is why documenting these designs is so difficult.
Swag beads can have straight lines running across them, making them essentially zone beads as well as swag beads. The best example I’ve seen of this is a swag bead with a reticella line running round the middle of the bead.
Another example of a bead having multiple designs are double swag beads with eyes inside the openings created by the two swag lines. These also tend to complicate the eyes by making them reticella or having eyes that look like rayed suns. A lot of these tend to be found in the British Isles, and there are some really nice examples in Scotland.
Callmer, J. 2003. Scandinavian Beads, ca. AD 700-1100. In: Glover, I., Hughes Brock, H. and Henderson, J. (eds), Ornaments from the Past: Bead Studies After Beck. London, The Bead Study Trust, 38-46.
Guido, M. 1978. The Glass Beads of the Prehistoric and Roman Periods in Britain and Ireland. London: Society of Antiquaries of London.
Guido, M. 1999. The Glass Beads of Anglo-Saxon England c. AD 400-700. Suffolk, The Boydell Press.