I'm 8 years sober, a nontrad postbac premed, and I'm leaving it out of my personal statement. It's a can of worms. I'd love to be able to share what I've learned in rehab and in AA, I'd love to let them see that part of me, the amazing tools of acceptance and gratitude, how much of the world's underbelly I've met and held hands with, how easy it is to put myself in another guy's shoes now. I honestly think my recovery, more than anything else, is what's going to make me a good doctor.
But people not in recovery don't get recovery. There are some anonymous eyeballs looking at the personal statement, scanning for red flags, scanning for great stories. Totally up in the air whether it's a red flag or a great story to these eyeballs. Then you (hopefully) get about 30 minutes of interview time where you get to sell yourself, and that's not enough time to even get started with the uninitiated about alcoholism and how you're handling it. And if it's in the personal statement, you pretty much have to be amazing talking about it. Total crapshoot.
If you're working a good program, and doing a good job of being a premed, you'll have enough other relevant experiences to set you apart. There are plenty of other applicants who have a bad patch on their record that has no better explanation than "partied too much." You have to have a fantastic list of what you learned and how it made you a better person.
I'm not worried about the adcoms who would think I'm another DUI waiting to happen - they wouldn't waste time interviewing me. I'm worried about the adcoms who would want my story to be just like a Lifetime movie with Melissa Gilbert and Meredith Baxter, where there are tears and broken friendships and strong words, but that's all over now, we can go back to being pretty. It ain't pretty, there's no cure, and yes I do hope to do AA my whole life.
Best of luck to you.