It’s completely up to you. Up until the 1940’s, many couples only placed a ring on the bride. (This tradition continues, most notably with Prince William and Kate, where he did not receive a wedding ring as part of the ceremony.) In some cases, one or both of the couple cannot wear a ring for safety reasons. Or they may just not feel comfortable with wearing any kind of jewelry.
I’ve also had cases where the rings were left in the safe at home, 3,000 miles away. No need to worry, your professional Wedding Officiant will figure out a solution for you. (I’ve used everything from cigar bands to twist ties, or just removed the ring portion from the ceremony. Nobody attending seemed to notice.)