That's a really interesting question, but the answer is no. You an rearrange your fragments in your halo between missions or sacrifice any to make room for more, but the only way another saint would get a memory you already had would be if it was randomly picked for the fragment draft when you win a mission. And, as you know, fragments come from the box, not other players.
OH, and I was thinking that maybe you could give a card from your halo to another saint on the Transfer phase, but the rule says "give a card from your hand to another saint at your nexus." So, you may only give cards from your hand unless a card says otherwise.