Just each Round when you know people need to prep Moves, get them to send it to you. You don't even have to really look at them, but it helps people not cheat themselves. Also, they're not consecutive! It's just a pool of options.
For NPC Moves I just write them down on a slip of paper beside my monitor and cross them off when I use them, I don't bother sharing them. I guess you could send them to everyone else not involved in that particular fight to heighten the tension? That could work. Just make sure to write up the NPC Moves before the game even starts, to make sure it's not a game about the fighting player trying to outguess you specifically. Keeps you honest.