The following is based on the Core pg24 and has been updated from the the Dark Heresy 2e Errata/FAQ, that addressed this subject.

• Whoever succeeds on their Test wins the Opposed Test. If both parties succeed, the party with the most degrees of success wins the test. If the number of successes is equal, then the party with the highest characteristic bonus wins. If there is still a tie, then whomever had the lowest die roll wins.
• Typically, the winner of the Test uses the degrees of success from his test result as the DoS for the test. However, in special situations, as determined by the GM, the DoS could instead be the difference between the two succeeding tests.
• If only one participant succeeded on their Test, then the number of DoS he achieved on his roll is also his number of DoS for the Opposed Roll.
• If it’s of any consequence, DoF is determined by the DoF the loser got on his roll alone.