Those in Alpha are technically under NDA, but Id imagine it is mostly in place to stop streamers. This game is also a little weird as they had an open "beta" for a week and a weekend for most anyone that signed up and everyone who preordered.
In the beta, weapons had there own XP system, that only leveled up on kills, if you hit the mob with that weapon, before it died. This is separate from the leveling system for your character (that gains XP from doing quests, mob kills, and tradeskill stuff). It is entirely possible to get to max level without skilling up any weapons (other than during the tutorial).
The weapons all have their own skill tree with six active and (a bunch of) passive abilities, I dont know if you can max it out, but you can only put 3 of the 6 active abilities on your bar(for that weapon). AFAIK, you dont get any extra passive dmg per point you put in the weapon tree, but each tier is locked until you invest a point in the preceding tier.
Here is a pic from the beta (not under a NDA) (not much has changed, at least for how the trees looks):