Yes you can transfer your costumes to another player as long as it is on the same account and on the same server as the one with the costume. Also the costume must be in the archive to be able to switch it to another character.
To get to the archive press L to open your quest log then at the top there are 5 tabs Quests, Fame, History, Archive, Rankings. You want to click the archive tab then go to other to see all the costumes able to be archived.
If the costume you wish to archive is there just drag the costume item that's in your bag into the little box beside the add button and press add.
Now your costume is in the archive, so what you need to do is log out and go onto your other character. Then open up the archive, go to the item you put in the archive and then right click it and it will go into your bag.
Step 4: Open your bag and right click the item to equip it.