A witness “without sin” is a credible witness in God’s law.
When Jesus asks for a witness “without sin” to go ahead and stone the woman caught in adultery, he is applying the law of Moses, specifically Exodus 23:1, “Thou shalt not raise a false report: put not thine hand with the wicked to be an unrighteous witness,” and Deuteronomy 17:7 “The hands of the witnesses shall be first upon him to put him to death, and afterward the hands of all the people. So thou shalt put the evil away from among you.” The accusers recognized they were wicked, unrighteous witnesses. In fact, if one flips through the translations, “without sin,” anamartetos in the Greek, is also used to translate the Hebrew chamas (evil, violent, wicked, cruel, and unjust) in the Septuagint. Thus, “without sin” is kind of too vague of a translation for what Jesus is really saying to the accusers, but the good news is that the law leads them to repentance, and they leave.
Once they’re gone, Jesus did not accuse the woman of adultery because under the law of Moses, he couldn’t: “One witness shall not rise up against a man for any iniquity, or for any sin, in any sin that he sinneth: at the mouth of two witnesses, or at the mouth of three witnesses, shall the matter be established” (KJV Deut. 19.15). He wasn’t an eye witness, and he certainly wasn’t going to set himself up as a false witness to the act; that would have been sin! There was no case against her once all of her accusers left. Although he did not condemn her, he did recognize that she was in fact guilty when he told her to “go and sin no more.”
This passage is historically misunderstood and even sometimes creatively misread. I believe the passage is really not about the power of Jesus to forgive (although that power is wonderful); it’s about the power of the law to reveal sin and provide grace at the same time.
Kehilat T’nuvah, the Harvest, has a podcast called “The Woman Caught in the Adultery Trap” in its Torah Talk files addressing this issue at some length.