Debeard or purge mussels first?
Soak first, then debeard.
I'm not sure that there is any real difference but everyone I've worked with, from grandmother to chef, from fisherman to fireman has done it that way. And there is something to be said for an amassed amount of experience.
One last to add though (and this is why I say that I'm not sure there is any real difference) is that while I get most of my mussels diving out with friends (so wild-caught and with beards and frequently barnacles to clean), most that I've seen in the past few years are commercially grown on ropes and usually don't have beards at all.