I try to discipline my kids exactly the safe way in public as in private. High but reasonable expectations for behavior. Calm, clear instructions. Follow-through on reasonable consequences. But, damn, it's hard.
First of all, reasonable expectations mean that, sometimes, my kid is doing something that I allow simply because of his development, but I get looks from others. For example, in a restaurant, at some point Loki (almost 2) can get down and walk around. To be fair to others, he certainly cannot go up to people at other tables or get in the waiter's way. But he can walk near the table, after the main meal has ended, while grownups have dessert, coffee, or chit chat. Developmentally, he can only stay in the chair so long. But, and I'm guessing this is because other patrons has experienced little kids disturbing their meals and think I'll let my kids do the same, other customers seem to glare, get uncomfortable, or feel they must entertain him (which totally undermines the whole "don't go by other tables" rules). I think it's absolutely appropriate. He's not screaming, touching others, or unsafe. He can explore, but only in our little area. So I do the same as I would alone. But it's so hard when others are glaring. :/
I also use timeout. With Ander, generally people compliment me. "Such a good boy." "I can't believe he stayed in timeout and then changed the behavior, but he did." "You are so calm and patient with him." But, with Loki, people inevitably think he's too little. They either smile and tease him to cheer him up while he is in timeout (which totally undermines the timeout) or they comment how young he is (though he is almost two and has been practicing in 10 second and up intervals until, now, he knows what is expected). No matter the good result, they don't WANT him to be in trouble, so they are uncomfortable. Also, the public didn't see that I simply redirect when possible (like in situations where no one else is getting hurt and there is no danger to Loki), use tons of positive reinforcement, and don't yell at or hit my kids.
I try to make yelling are rare occurence with my children. I really do, But not yelling at other people for their reactions to my discipline techniques, which are kind, developmentally-appropriate, and, dammit, WORKING - that is a challenge.