I am and have always been a private person. But I also live in a small town. After several unsuccessful attempts at getting sober in secret, I did it very publicly the last time I tried. And it stuck. I don't feel like I ever would have gotten sober if I didn't "wear it" and identify as someone who no longer drank. Other people's reactions ran the gamut and it was important for me to be able to process that and understand that other people's opinions had no bearing on whether or not I remained sober. Being secretive and ashamed about it never helped me.
But I definitely agree that it's a personal decision.