No, you can't have private posts or posts that are only actionable for a specific user/group of users on a public StackExchange site, say AskUbuntu.com.
You either need a subscription for SO for Teams (note that the teams option only exist under Stack Overflow) or get a license for SO Enterprise (but that is only beneficial if you have lots of private users and enough money to burn).
If you already have an Enterprise instance you can add Teams on top of it. That would give you private teams within your own Enterprise instance. Your Enterprise Customer Success representative knows the details about this option and can assist in enabling this for you.
Both Teams and Enterprise have some limited ways to interact with it by the use of an API, and/or Slack integration, allowing for internal routing of questions.