Unlike Stitching the original items (in this case pets) are NOT removed from your inventory. If you mix two of your own pets, you get to keep the original pets PLUS you get the egg they create. If you mix with a friend, your friend keeps their original pet, you keep your original pet and you BOTH get an egg.
Your egg only hatches with time, young Wizard. Inspect the egg in your backpack, there will be a countdown timer to indicate how long until it hatches.