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2.
The "two-backed beast" is a euphamism for having sex. For some reason, people can't just say "having sex," they have to come up with a clever way to say it.
"Last night I was making the two-backed beast with my boyfriend"

Bumping uglies, knocking boots, burying the salami, etc
by Asty von Ferguson October 06, 2005
 
1.
N. Old literary term for sexual copulation. The most famous being a quote from William Shakespeare's Othello where Iago informs the King that his daughter has been having sex with a black man. Though modern high school students generally assume that making the two back beast is a reference to doing it doggy style the literal meaning in the 15th century was sex face to face or in the missionary position.
"If you tell Momma I was spying on you, then I'll tell her you and Jorge were making the two backed beast."

William Shakespeare. Othello. (1605): Iago: ' I am one, sir, that comes to tell you, your daughter and the Moor are now making-the-beast-with-two-backs .'
by Keifermail July 27, 2008