To make a recipe private/public, visit the recipe page and select the three dots located at the top of the screen > "More Options..." > "Edit Recipe." From here, depending on the status of your recipe, you can either mark it as "Keep this private" or "Make public." Once you're finished, scroll to the bottom of the screen and hit the "Final Check and Submit >>" button.
By default, all clipped recipes are made public -- this is because more than one user can import the same recipe. Conversely, RecipeScans are uploaded as private. Users can set the status of a typed in recipe when adding the recipe -- simply check or uncheck the "Keep this recipe private" box.