There are several questions here - so let me comment on them briefly. (This answer is mostly gathering things already mentioned in the comments.)
A very short answer to the question from the title is: If by "private" you mean something visible only to you and to the other user, there isn't such an option (unless you become a moderator). Some related feature requests: Should we request a Private Message system? and
Any way to send a personal message to another user?
Various types of rooms
- Only moderators can create private rooms - the messages posted there are available only to the moderators and to the users that have been explicitly granted access to that room.
- Even regular users can create gallery rooms - the conversation is visible to everybody, but the room owner(s) can specify who is allowed to talk. To be able to create such a room you need at least 1k reputation points.
- Regular rooms (with no restrictions) can be created by a user who has at least 100 reputation points. Keep in mind that you can choose various parent sites when creating a room.
- One can create a one-to-one room - using the button "start a new room with a userXYZ" in chat profile. They are basically the same as the normal rooms - but in the list of rooms they are located at the end (even after the rooms imported from comments).
See also: Is there a list of SE chat privileges, and the minimum reputation required for those privileges?
Inviting user to a chatroom
You asked how to invite a user to a room - one can find the information here:
Or you could simply ping the user in a comment - and include a link to the chatroom in that comment.