Another thing some people do is that, after the groom gives the bride the ring and says, “Harei at mikudeshet li c’da’at Moshe v’Yisrael” (Behold, you are set apart for me according to the laws of Moses and Israel), the bride then says, “Harei, atah mikudesh li c’da’at Rebbeinu Gershom v’Yisrael” (Behold, you are set apart for me according to the laws of Rebbeinu Gershom and Israel.)
Rebbeinu Gershom is, among other things, famous for issuing a ban (“takanah”) on polygamy in the 11th century. At first it was understood as binding on specific Ashkenazi communities, but now it’s more or less accepted accross the Jewish world (with a few communities here and there who don’t hold by it). As such, for Ashkenazi Jews in particular, Rebbeinu Gershom acted as the agent of more or less equalizing the male and female pieces of the marriage agreement–just as the bride, through marriage, was prohibited to other men, the groom, through marriage, is prohibited from taking other wives. (The issue of adultery is complicated, and I won’t get into it here).
The big problem with a woman handing a man a ring and saying, “Harei, atah.. c’da’at Rebbeinu Gershom” is that it’s technically not correct. The takanah (forbidding him to marry other women) kicks in when he acquires her, NOT when she gives him a ring. It may be a nice, symbolic way to acknowledge their respective sexual limitations on each other, but it has no halakhic status.
ADVANTAGES: Halakhic marriage, ring exchange, the woman publicly declares her husband’s monogamy just as he declares hers, it’s kind of cute.
DISADVANTAGES: Her statement and ring-giving have no halakhic status, she’s still acquired (and he’s not, really), even the limitations on seuxality in the two situations aren’t exactly parallel.