The primary activity of the right brain is learning and using all the connections between all the elements of our world. These elements include all aspects of the natural world, all the constructions of man, all thoughts and feelings in our minds, and most importantly all the other people we interact with.
We are social animals, much more than we normally admit. The only other animals that live anywhere near as densely as we do are ants, bees, and termites, and we call them social insects. We have conflicts, personal, familial, and national, but most of the time in most places we live together, often very close together, leading productive and fulfilling lives.
That's the right brain. The right brain sees everything connected, including all the human beings it perceives. You can't escape that socialization, though you can choose not to participate at various levels. However much you choose to live apart, your life will be defined by that separation, and much of your right brain activity will be mediating that apartness.
The right brain has mirror neurons (watch this TED video: it's fascinating.) This is really a fully integrated major subsystem that runs through much of the right brain, though it is mediated in the frontal cortex. Mirror neurons allow us to fully understand what another person is doing physically. When we see another person lift her hand, the same neurons fire in our brain that fire when we lift our own hand. Likewise, when we see another person hurt her hand, the same neurons fire in our brain that fire when we hurt our own hand in the same way. (See the video for the reason we don't feel the same pain.)
VS Ramachandran calls mirror neurons "The neurons that shaped civilization". I call them the neurons that make us human. Just about every other neuronal system has an analog in other animals. So far we have only found limited mirror neuron activity in non-humans, whereas we are immersed in it.
Every second of our lives in which we are perceiving others, we are partially living their lives. Our right brains take in every nuance of position, action, and reaction that the people around us present. And they are doing the same with us.
This detailed and intimate observation allows us feed our theory of mind. Theory of mind is not a psychological theory, per se, but a theory my mind holds about the content of another's mind. I infer what the other person is thinking from the location and observed actions of the other, that is, my mind has a theory about the content of the other's mind. Because the right brain is grounded in reality, it doesn't turn the theory of mind into "fact of mind". It remains a theory, which is continually updated from further "observation". Note that "observation" here means the intimate feeling of "mirror neurons", not rational seeing and thinking about it. All unconscious in the right brain without effort, or even choice, unless we divert our eyes.
One of the most powerful aspects of mirror neurons is the observation of other people's eyes. Our right brains typically maintain a 3D picture of where we are in the world. The picture is so accurate and detailed that we can often correctly infer what another person is looking at by seeing where they are and where their eyes are pointing, even if the thing they are looking at is not visible to us. This is an incredibly complex geometric calculation that our brains can perform in real time while doing other things.
One of the most notable aspects of mirror neurons is the observation of other people's faces. Our minds express themselves richly in our faces. Such expression is very subtle, and there may be dozens of aspects of our faces that embody that expression. Our mirror neurons allow us to infer what thoughts we might have been thinking that created the exact movement of a specific facial muscle that we see on someone else's face. To the extent that we are right, we gain insight into the person we are looking at. We can also be wrong since we express ourselves somewhat differently.
The target of mirror neurons extends outside the body. When we see a car driving down the road, we not only infer the state of the car but that of the driver inside, even if we can't see the driver. And we do this for many cars all at the same time. The amount we can infer about the state of the other drivers' minds is limited, but it is sufficient to allow us to safely drive in crowded cities and freeways.
Our babies are born into the secret world. They are born with the skills to participate in it immediately. They don't recognize a boundary between self and other. There is only the secret world and all the wonders in it. Parents who bond with their babies are being re-initiated into the secret world. As the baby becomes initiated into the constructed world, learning words and concepts and proper behavior, the child is often forced to choose which world to attend to. The secret world never goes away, but most of us, parent and child alike, lose touch with it as the child grows up.
It seems to me that a pack of canines shares a similar secret world communicated (mostly) non-verbally. I don't know the exact nature of that secret world, but I suspect is similar to our own. (For example, a dog can do the same 3D positioning trick I described above.) I believe the reason dogs have been such successful human companions is that we can share some part of our secret worlds.
Whether we are walking down a crowded street or sitting at home with family, we are participating in a continual, omnidirectional communication with those around us, and the knowledge we infer from this is as real to us as any other sensory input.
The secret world is always there, but we are not often consciously aware of it. The experiences and circumstances of our lives often teach us to disconnect our consciousness from the secret world. There is a lot of emotional content shared in the secret world, and much of it is sad, and some of it is disturbing, so we learn to ignore it. There are people who can't hide from the secret world, and there are many who never experience it directly. But we all use the information from the secret world to allow us to live in (more-or-less) harmony with others.