At my school, which had 8 sororities (7 now) and about 300 women going through recruitment, almost every woman matches, so fortunately the rho chis never had to deal with telling women they didn't get a bid. Unless they suicided, which several women did every year. We called those women ahead of time so they knew not to come down to collect a bid.
For the rushees that got bids, each rho chi was in charge of handing out their rushees' bids. We usually had a time frame in which our rushees could come get their bids; most women would show up right away, but some women that were nervous would actually wait until the last minute. Each rho chi group had a table in an area of a nice, private room, and we would hand out bids to the women as they showed up. Some women would tear open the bids right away, in front of everyone, whereas others would take the envelope and go into the bathroom or outside to open it privately. It was never a frenzy, and I don't recall a lot of crying, either. Note that we did not have an option to accept or deny a bid; we were told by NPC after preference parties that by listing sororities on our final bid preference form we were indicating we would accept a bid from any of them.
Once bids are handed out, all the new members and rho chis gather in the amphitheatre for the rho chi reveal. When I was there, all the rho chis would sing a portion of a rush song/chant from each chapter on campus, then on the second round of singing the medley, they'd remove their rho chi sweatshirts to reveal their GLO letters underneath (at the appropriate time, of course
) Then the rho chis gather up the new members of their sororities and lead them off to meet their chapters, who wait for them in assorted nearby locations on campus.