To send someone a gift, go to Gift, search for them by username, and pick an item to send. The cost comes out of your own diamond balance, not theirs — you're the one paying for the gift, and it lands directly in their Inventory once sent.
If you don't have enough diamonds for the item you want to send, you'll see how many more you need before you can complete the gift. This is also how sending someone an item from their wishlist works — tapping a wishlist item on their profile takes you straight into this same gifting flow.