Private graphs are only available to accounts that have purchased a developer, team, or enterprise plan.
Saving a private graph
When saving a graph, you will be able to determine your sharing preference - whether you want to make a public graph or a private one. Note that once you have a paid plan, by default your sharing preference is set to private.
Note that users with a developer plan can make private graphs that are hidden from all other users. Users with a team plan can make private graphs from the public, but still share their private graphs with their team. Developer plan users cannot share their private graphs with other users.