When you reach the Great Buddy rank with your chosen Buddy Pokémon in Pokémon GO, you have the chance to receive presents from them. Make sure you have the second heart underneath their name when you view their Buddy profile before you attempt to go after gifts.
Your Buddy will randomly find these presents while you’re out exploring the world. To make sure they have the chance to find them, you want to make sure their stomach is full in the lower-left corner of your screen. If it’s empty, you won’t see them flying around your Trainer Avatar, and you’ll need to feed them berries.
You’ll see a small wrapped gift underneath their Buddy avatar on the Buddy Profile page. To receive the present, you have to “Play” with them in the AR mode.
When you first reach Great Buddy status, your Pokémon should have one immediately available for you to unwrap. The items inside of these presents vary between berries and useful potions or even revives.
Many Redditors believe you can only receive one present from your Buddy per day, much like another friend only being able to send you one per day. If you have multiple Pokémon with the Great Buddy rank or higher, you might be able to receive numerous presents because you can receive 20 gifts per day from all of your friends. The system may work the same way, but this is unconfirmed.