Ever heard that expression, 'Innocent until proven guilty'? Well, that simply means an assumption of innocence until enough evidence can be established to prove guilt. As for the expression 'born sin-free' that is not a Christian concept because the Bible is clear in showing that every person is born into sin and every person will sin. Otherwise, we would be perfect. Only Jesus was without sin, and Adam and Eve were without sin prior to their act of disobedience.
My understanding of Jehovah's Witness' theology on sin and salvation is they believe Adam was created perfect, but when he sinned, he lost eternal life for all of his descendants. They say Adam chose to sin, thereby giving up his perfection but we, as his descendants, were born imperfect but not through choice. They believe that Jesus (as a perfect man) bought back what Adam lost - the right to eternal life. As for our inherited (or Adamic) sin, they say that because the wages of sin is death, then when we die, we pay for our own sins. Jesus' sacrifice, they say, pays only for the sin of Adam.
"Adam forfeited his own perfect human life and the right to eternal life in exchange for selfish use of his free will. In so doing, he sold all of his future offspring - including us - into sin and death. Our opportunity for eternal life was lost... our human race needed a redemption to ransom them from sin and death and to restore their birthright... Christ forfeited his own perfect human life and earthly prospects to purchase the life rights of our own human race." Victory Over Death - Is It Possible For You? published 1986.
I cannot say whether Jehovah's Witnesses believe that whilst being born into sin we are innocent (because we didn't chose to sin, unlike Adam), but I can say that the suggestion Jesus' death was only to atone for our supposed inherited rights to everlasting life on a paradise earth can not be substantiated from the Bible. 1 John 2:1-2 clearly states that Jesus Christ "is the atoning sacrifice for OUR SINS, and not only for ours but also for the SINS OF THE WHOLE WORLD."
The following link discusses in depth the subject of Adam and of sin and may be of interest to Jehovah's Witnesses: http://home.sprynet.com/~pabco/adamsin.htm