No and yes. 1. "I don't owe you anything. You don't owe me anything either." In this case, the first person [I and me] is talking to and about the second person [you]. 2." I don't owe him anything. Neither do you." In this case, the first person [I] is talking to the second person [you] about a third person [him]. So, because of the pronouns you used, 1 and 2 do not mean the same thing. However, in your second example, they do mean the same thing. 1. I don't owe him anything. You don't owe him anything either. 2. I don't owe him anything. Neither do you.

In the first set of two sentences, the first sentence has only two people; let's say Tom and Jim. (Tom doesn't owe Jim anything, and Jim doesn't owe Tom anything, either.) However, the second sentence has a third person (him). The word "him" refers to a third person, Bill. For example, Tom is talking to Jim and says, "I don't owe Bill anything, and neither do you." Neither Tom nor Jim owe Bill anything. The second set of 2 sentences have the same meaning.