When you buy Don’t Starve Together on Steam, the first copy goes straight to your game library (as it normally does), and the second giftable copy goes to your Steam inventory. You can get to your Steam inventory by hovering your mouse over your name at the top of Steam and choosing "Inventory". It will be in the Steam tab.
On the other hand, if you purchased Don't Starve Together, chose "as a gift" (or if you already own the game and buy the game again by choosing the option “as a gift”) and sent the gift to a friend, the person who receives the gift will automatically get Don't Starve Together activated on their Steam account and game library (as it normally does). That same person will also find a second, giftable copy of Don't Starve Together in their Steam Inventory that they can send to a friend.
Visit the Steam Support site for more information on Steam Gifts.
If you encounter any issues with your purchase or with your giftable copy, your best bet is to contact Steam Support directly. Steam Support will be able to check your account/purchase and provide assistance promptly.