Once upon a time, there was only one ring; the best man gave it to the groom, who gave it to the bride who wore it to her grave.
The groom didn't wear a ring at all, and the whole arrangement was rather one-sided, and redolent of ownership, possession and control.
As those ideas went out of fashion during the latter half of the 20th century, men and women began to exchange rings, but for some reason, the best man remained the custodian of both, which gave rise to the strange sight of a bride getting a ring from the groom's best pal only to give it straight back to the groom.
It took me a little while to see what was wrong with that picture, but now I have a solution, and most couples seem to agree it's a good one.
Rather than making the best man responsible for both rings, I usually suggest that while he has the bride's ring to give to the groom, the chief bridesmaid should be given the groom's wedding ring to give to the bride.
That way the exchange is truly equal.
I generally suggest the bridesmaid simply wears the groom's wedding ring on one of her thumbs. As both thumbs tend to be pointing upwards (because they're carrying a bouquet), there's no risk of it falling off, and it's a simple matter for the bridesmaid to slip the ring off her thumb, and pass it to the bride at the appropriate moment.