Speaking as a person who hates talking about my emotions, and hates telling people the deepest personal stories of my life, the best way to get a person to open up is to shut up and listen. A person who is afraid to share is afraid because they don't know how you will react, or if you can be trusted. They are afraid if they tell you who they really are, they will be rejected. You have to show, through time and consistent friend actions, that you are open to what they have to say; that you will not judge or judge negatively the things they think or have to say; that it is safe for them to bring their darlings out into the open, and that you will not smash them. Don't try to pry out of them what they don't want to talk about. But do be open with your own secrets and vulnerable spots. When you are vulnerable in front of someone else, it makes them feel more safe: you have risked something, you have put your skin in the game, which means you trust them enough to tell them something which could hurt you, and it hasn't, and maybe in return they can begin to do the same.
As someone with a similar personality who learned this the hard way, then had this reinforced more times than I want to account for through therapy groups and training for 7+ years, this is spot on. One bit of knowledge I can pitch in here as well is the use of this particular dialogue to get into the habit of 'active listening'. It's clumsy at first, but provides some seriously great training wheels for creating a safe space for 1-on-1 opening up to one another. That said, it's based on a model for 'couples therapy,' which can be applied to any form of relationship, in reality. My favorite part of it all is how commonplace the wording can be, and the efficacy around de-stigmatizing opening up is profound in my experience. Your partner in these interactions may soon follow your lead and consciously, or not, start reflecting back and listening - a win-win in my book.
Thank you very much, ref, for a good advice. I must admit not to be patient most of the time, and with people, I see that it might get... difficult, for lack of a better word. I also steer the conversation towards self-imporvement a lot of time, none of which has been called or asked for yet. In my trying to help, I might be intimidating the trust of the person I want trusting me. The intent is most benevolent; I'm afraid that the result isn't always beneficial. I'll do my best in that area and see how things go.