Private sessions are an opportunity to bring your work with the Enneagram to a deeper level. The Enneagram can be nuanced and complicated, and different teachers will have different approaches, so it can be tricky to parse out what will be most relevant to you personally. So one component of private sessions is how to make this knowledge immediately useful to you and the life you have.
Private sessions are also, like therapy, spiritual guidance, and coaching, a way to work through personal blocks and discover a more fulfilling way to live. The precision of the Enneagram offers a uniquely powerful way to address the struggles we face in life: knowing somebody is a Six rather than an Eight, for example, can provide important insight as to what will be most useful to look at in someone's journey.
The way I use private sessions with my own guides is to dig into questions of meaning, to engage in shadow work, to work through my family history and how I approach relationships, to practice being present to my reactivity, and to cultivate a meaningful relationship with my own inner life--or whatever else arises in the moment!