I don't think you really ever stop fear. Heck I am just as fearful of dying as I was before I had all my training. Maybe even more so. I think you learn through training how to drive through fear by developing coping mechanisms and habits to allow you to push through it.
It's not that you have no fear... it's that you stop being fearful of as many things. I have a number of friends who are combat veterans, very hard core. Some were really damaged by the experience. But a number of them, especially the guys who had a personal practice like the martial arts, simply ended up as guys who are not scared by the million and one things that effect most folks.
I think that physical bravery is the easiest trait to develop. Just look at the extraordinary acts of courage done by ordinary people put into combat situations or natural disasters. But look at how difficult it seems to be for human beings to stop being afraid of each other...
I have friends who wouldn't bat an eye being on the mat with five guys with sticks trying to hit them... but ask one of them to have a sincere conversation with one of his peers? You'd think it was asking the impossible. The toughest guy you've ever met will let his marriage crumble rather than go to counseling. The prospect of talking about his own feelings is just too frightening.
People are terrified of being hurt by each other. That hurt can take the form of criticism or judgment. It can take the form of rejection. In the extreme you get a young man who shoots another because he was "dissed". Virtually every way in which we are fearful of our fellows produces a way in which we can hurt them. It goes back and forth and the result is a society of people who are terrified of each other. Then, when the powers that be go out of their way to exploit this fearfulness for their own ends, you end up with the kind of crazed, polarized mess we have today.
Aikido training should develop ones awareness of the fundamental connection between all of us. You have to be willing to put oneself into a physically intimate relationship with your classmates. You need to make yourself vulnerable, just as in relationship. There can be no connection, no technique without being vulnerable. Contrary to what many folks who are "fighters" might believe about the art, I think that one of the most important aspects of our training is learning to "lose" i.e. take the fall, receive the technique.
In life we "lose" all the time. My wife dropped divorce papers on me, a student lost his job, a boss rains all over you about something that wasn't your fault, a child is killed in some war far away, a hurricane destroys your home, a fire burns your dojo to the ground, it is endless.
There is no magic technique that keeps one safe from these things. What? You're going to nikkyo the boss when he's being a jerk? What cool fighting technique will protect you from the devastation of a child passing? No, you are taking a hit. After that, it's a matter of how you handle it, what kind of ukemi you take. If you contract around the pain and hold onto it, the hit may be so hard it destroys you. Or you can go with it, let it move through you, and perhaps take it into something more positive.
I believe that Aikido practice, when it done well develops the ability to stand at the center of chaos and be strong, physically and emotionally. At the same time it also helps one realize that the winning and losing model we often buy into in our lives, is simply not functional for most of our human experience. Shit happens. You aren't in control of it, you can't defeat it, resistance is futile. That's the ukemi side. Things are going to happen in which you are taking a fall. Do you want to hit hard and hold that injury in your life or can you take the fall and move on? As a lapel pin I once had said, "Live right, eat healthy, die anyway."
That's the central fact of existence. We are all going to die. Good martial arts training should heighten ones awareness of just how fragile the human being is. At some point, you realize that fighting is really a no win proposition most of the time. As we can see from our various military enterprises in my own lifetime, in doing what is necessary to "win" we end up damaging ourselves on a very core level. The price of such a "victory " will be paid for at least two generations in the damage done to the current participants and the the issues they pass on to their own kids because they haven't dealt with the damage.
It's not that Aikido is unique in this at all. Many of the most amazing, high quality individuals I have ever met are lifetime martial artists. But Aikido is specifically structured to develop this sense of connection coupled with a letting go of attachment to particular outcome. In this interaction with our partners we learn to relax and allow the partner to act as he or she wishes and let the technique become what it needs to. Take musu aiki. And if suddenly we are taking the fall, that's ok too.
While this paradigm might not be the best one for developing fighters for combat, I think it a very good one for developing human beings who can live their lives doing more good than harm, leaving things around them better than when they arrived in the world.
So Aikido can help people lose their fear of being intimate. It can help them to stop worrying about "losing" something when dealing with others. In short, it can help people move out of the "if I'm not winning, then I'm losing" mode of thinking most folks operate under.
So, it's not that we stop being afraid. We just narrow down what we are afraid of to what is of real significance and stop being afraid all the time of what doesn't actually matter. When we stop being afraid we gain our freedom to act.