Once upon a time, the rehearsal dinner was for the actual members of the wedding party and their spouses (if there were any). With a Saturday wedding, you would have a rehearsal on Friday night - making sure everyone knew who would stand where at what time, etc. Then after the rehearsal, you would go out for a nice dinner. This has expanded over time to include other out-of-town guests if the groom's family wants to include them, and family members not in the wedding, etc. Some of the rehearsal dinners I've been to have been a more elaborate meal than the wedding reception (just not cake or dancing). But again, it's always been the groom's family who hosts, not the bride's.
Just as a side note, when my sister got married, I was her maid of honor. Apparently there was a tradition there that it was bad luck for the bride to do the rehearsal, so she sat in the front pew and watched, and as the made of honor, I stood in for her at the rehearsal. That was interesting. I've never encountered that before. Another cute tradition is that at the bride's wedding shower, she saves the bows from the gifts, and makes them into a bunch that stands in as the bouquet of flowers at the rehearsal. Since you actually practice handing the bouquet to the maid of honor while you join hands for the actual ceremony and such, it makes sense to have something to practice with.