This is all hypothetical, and told, mostly from a submissive male's perspective, in the service of a beautiful dream goddess.
She led him through the mall, by the leash around his neck. Forcefully threw him against the nearby wall by the food court. "Face the wall." She commanded. "You will keep your EYES on ME at all times, and NEVER ogle those other girls! DONT THINK I DIDN'T NOTICE, SLAVE!" She barked. Then, reached into her purse for her paddle. WHACK! came the paddle, hard against his buttocks. He whimpered in pain. "I'm so sorry Mistress..."
Meanwhile, a mother and her two daughters scurried away, all three with looks of horror on their face. A young boy sitting at a nearby table said, "Why is that woman so mean? And why does that man have a leash on?" The father, aghast, did not even know where to begin to explain.
Then, another couple at a restaurant. The man, kneeling adoringly in front of his mistress. "Worship my feet. Thank you slave!" She barked. The man, in a daze of frustrated lust combined with blissful adoration, willfully complied, oblivious to the staring couples at the table across the room- and the waiter standing nearby. There was whispers and laughter. Then, the manager walked up, and politely but firmly asked both Mistress and slave to please leave the restaurant.
So, bottom line...these scenarios don't really work so well. In a public place, discretion is the better part of kink. Kids, and members of the general public, probably aren't really ready to be unwilling participants in a public humiliation/discipline scene.
But yet...there are scenes in public that CAN work. In a discrete way, of course. The sub/slave could wear a short leash, not visible around his neck but around his balls, with only a small loop visible sticking out through his zippered pants. As long as She is not blatant about it, an occasional firm tug on the short loop, in an uncrowded place, would still give him a clear message about who is in charge. Or, he could wear clothing and jewelry that clearly indicates his position relative to Her, which a BDSM practitioner would be able to identify but a casual, perhaps more innocent, observer may not. Even a T-shirt with a "PG-rated" saying ("Property of Ms. Julie," for example) might be okay. And what he wears UNDER his clothes, of course, is entirely up to Her wildest imagination.
There are some settings where it might be okay to push the envelope as far as public play. We're not talking Temple Square in downtown Salt Lake City, of course! In some cases, going too far with "Freaking out the Squares" could even get you in trouble with the cops! But rather, places where "adult play" is more socially accepted and- key point- there are no minors or family types around. For example, nude beaches, strip clubs, even some bars. Some ideas: Make him wear tiny slutty shorts, or even fully cross dress (in some bars or settings this might be acceptable.) Take him to a male strip club and openly flirt with the other men in front of him, and maybe buy him a lap dance and force him to enjoy it (and pay for it, too.) I have even heard of more extreme scenarios, planned out in advance of course, where She forces Her plaything to work a glory hole - and subsequently and submissively service one of her acquaintances, (for safety's sake it would have to be her acquaintance) Or, if ever She feels he needs to be corporeally punished, drag him to a private area, away from private eyes, and administer it where no family-with-kids types will see it. If there is a known "Kink-friendly" bar or play space around- so much the better! Then, it's, Mistress, do your worst... But sadly such places are few and far between in the heartland cities.
But the key thing is, there are ways that these public scenes can happen in ways that do not disrupt the "Squares," families and kids going about their daily lives. It's just a matter of the setting and the surroundings, and the way in which it is planned- Her wickedest imagination can know no bounds.