My wife asked me this over breakfast, and these are some of my thoughts on this question:
The first association that most people have, I guess, with regards to "an intimate conversation", is that it is either about sex/romance or about some pain/weakness that the person usually hides. But I'm more interested in a different type of intimacy. That type does have some of the erotic flavor and the trust, vulnerability and openness of these two, but it's different than them.
In my experience, an intimate conversation is one in which the boundaries between us somewhat dissolve or become permeable. To an extent we find ourselves thinking, feeling together, intuiting together new understandings.
Sometimes, the awareness of the "third entity", which forms between us, overrides our awareness of ourselves as separate individuals.
What enables such intimacy?
It is born out of the intention to discover and reveal something that I don't yet know, and to enable the other person to do the same. So I renounce a lot of what's readily available to me - stored information and opinions. Instead, I look into my experience and try to express what freshly comes out of it. If the other person does the same and we closely attune to each other, the miracle happens. The conversation slows down, the quality of listening deepens, and the space between us like becomes denser.
This for me is an intimate conversation.