The planner is inherently not designed for Private chats. Its underlying model makes it a service that is open and collaborative. Indeed, you can’t have a private planner, but you can always add the participants in a private chat to be added in a planner (not privately). Private channels were initially introduced to sensitive information within a Teams channel like budget, login credentials, resourcing, etc. Private channels can be composed of a subset of people within Microsoft Teams.
Since Private Channels are a new thing in Teams lately, according to Microsoft, Teams private channels do not support apps like Stream, Planner, and Forms. All other connectors and tabs are supported. Apps like Planner have a pretty tight integration to Office 365 groups and thus they are not yet supported in Teams.
Get in touch with a MachCloud Expert to know more details on Teams integration with other apps.