Commentary for Kiddushin 130:7

R. Kahana and R. Papa argue over whether we are concerned for kiddushin done in front of one witness in a case where the man and woman agree that there was kiddushin. R. Kahana says that such kiddushin are not valid. But R. Ashi argues with him. Perhaps R. Kahana is thinking that just as the word “davar” stated in reference to monetary matters (Deuteronomy 19:15) means that we require two witnesses, so too the word stated in reference to sexual matters (Deuteronomy 24:1) implies that we need two witnesses. The problem with this analogy is that in monetary matters, a person who admits he is liable is liable. But R. Kahana is arguing that even if the couple says they were betrothed, without two witnesses, the betrothal is invalid.
R. Kahana responds by noting the difference between the two situations. In the case of monetary matters, if a person admits he owes money, he is not causing others to be obligated. But here, the couple’s admission impacts each other’s relatives, who will not be able to marry them. Therefore, it is invalid.