Don’t know yet. I’m hoping it works just like normal gear. If you have it on a character that can use it, any other character on your account that can use that type of item can transmog it.
I have the bow on my Warrior but I’m hoping I can use it on my Hunter. (Although I don’t know if Warriors can even equip bows anymore, so I don’t know if that will even work. It’s just been in his bank since he got it.)