People have different perspectives on this. I guess there's a desire to broadcast something you're proud and happy about, but tempered by the reality that most people will never "get" addiction. They absolutely will never "get" relapse - "WHAT, after all you went through, you're going back to drinking/drugging, what is wrong with you???"
My friends and family know the story and status, but for the most part no one else does. If drinking comes up as a topic, like for a post-work bar trip, I'll go if I feel like yakking but I'll make it clear up-front that I don't drink. Many people don't drink, so it's not a big deal, and very few people have ever inquired as to why I don't drink. If they do, I might go as far as, "I already went through my lifetime quota of alcohol, no more for me", but not into details unless I sense they do understand addiction.
Once you broadcast something like that, you can never take it back, and you can't control how people will react or change their opinion of you, for the better or worse. To me, better to maintain some control by only discussing it with carefully selected people, and otherwise keep it as a personal matter.