Looks like the proper replacement to the now-defunct Swapnote.
Contains in-app purchases but there's a free starter pack to get you going:
Starter Pack: Includes four lessons, two types of stationery and one coloured pen. Increases maximum note storage by 100 and ink capacity by two units.
Share personalised handwritten notes, drawings and more with Swapdoodle, a free-to-start* messaging application only for Nintendo 3DS family systems. By registering friends and family as Doodle Pals, you can exchange all sorts of customised messages over the internet**. You can even learn how to sketch familiar Nintendo characters with purchasable doodle lessons.