All three sentences are grammatically correct, but they have slightly different meanings. Here are some dialogue examples to illustrate:
A: "I don't owe you anything."
B: "And I don't owe you anything either."
A: "That's fair. We two owe nothing to each."
A: "I don't owe you anything."
B: "Well, I don't owe you anything either."
A: "I guess that's true. We two owe nothing to either."
A: "I don't owe you anything."
B: "And I don't owe you anything either."
A: "So we're even. We two owe nothing to each other."