Check Winner DLC

you can't..

Steam generally doesn't list them in the library.

If it is an inventory item, send the user a friend request. Once they have accepted, go to your inventory and send it to the user. If they own the DLC, Steam will show it.

With a key, well you can't check unfortunately. :(

You can check, with most stuff. Send a friend invite and then try to gift DIRECTLY through steam. Not to email. The list it shows you SHOULD mark whether they own it or not.

I wouldn't suggest checking the Steam Store pages that others have mentioned as it does not work every time. The store doesn't show that I or anyone else on my list owns these DLC despite them being included in Dead Island GOTY.

http://store.steampowered.com/app/201741/
http://store.steampowered.com/app/91346/

If you add them and send the gift that way, it does indicate which people already own Dead Island GOTY and even those who only have Dead Island. As such, this method is obviously much more effective.

As Rinarin said, you can check if anyone on your friend's list has any game, even if their profile is private.

Add the game to cart, select "Purchase as a Gift", then do "Send my gift directly through Steam". It will display your friends and it will have a note if the person owns the game.

Then delete it from your shopping cart.