yes they do stand like that...girls on one side, boys on the other, facing each other.
This is just their starting position though.
There's a person that's the "caller" and he/she leads them in the dance.
Usually with the phrase, "Bow to your partners" where the boys bow and the girls curtsy to their partner.
Where are you from? I ask since square dancing came to America from Europe, lots of different version.
below are square dancing costumes...the women where full skirts because they twirl around a lot, making the skirt fan out.