I feel strongly that there is a huge difference between honesty and privacy. And I think that sobriety falls into the privacy category.
What's funny is that now that I'm sober I'm totally honest with my work. Late because of a flat? Honest. Ran out the door and forgot my work keys? Honest. Sick in bed with the flu? Honest. I'm an open book now that I'm sober. I lied all the time when I was drinking. People would ask "Are you okay?" I'd lie and say I had trouble sleeping. Late because of a hangover and keys were lost when I was drunk? I lied and made up a story. Why am I sweating in January? Must be coming down with a bug (again). Ah crap, used that one last week. Ummm, I just ran up the stairs. My eyes were also red because my contacts were bothering me. Lie again.
I'm pretty darn honest now but not about my sobriety though. I am honest about being a non-drinker even though you'd be astounded at how rarely this actually comes up. One of my favorite things about being a non-drinker is that none of my behaviors outside of work affect my work one bit, one shred, one iota. Work party? Sure, people know I'm a non-drinker. So are 30% of people.