Some confusion arises sometimes when people discuss reward-based training because it can be hard to understand how you teach a dog not to do something without making it unpleasant for him. People who don't understand—or who don't like—non-aversive training sometimes confuse positivity for permissiveness, not understanding that rewarding what you want while ignoring what you don't want is not equivalent to letting a dog do whatever he pleases. Positive does not equal permissive. You can be quite structured with your dog and hold him to a very high standard of reliability and precision without scaring him, intimidating him, or inflicting an unpleasant sensation on him.
However, you do have to be careful when you ignore undesired behaviors, since some unwanted behaviors allow dogs to reward themselves. It might be more helpful to borrow an awkward word from psychology and suggest that trainers get in the mindset of nonrewarding undesired behaviors rather than ignoring them.
For example, if your dog is chewing on the drywall, you can't just ignore him. He is rewarding himself with the chewing. If your dog steals a cookie from your toddler, eating the cookie is a strong reward for the behavior, so ignoring it is essentially rewarding it. In both scenarios, the dog is being rewarded and thus is learning to do the behavior more frequently and more intensely.
If, however, you nonreward these behaviors, they will diminish, especially if you reward something incompatible with the undesired behavior. If you know your dog wants to chew the drywall, you need to manage the problem by removing his access. A baby gate, a piece of furniture, even a tether to you could help you make sure he's not rewarded. And you should reward him for taking out his chewing urges on appropriate toys. When he chooses something like a durable Nylabone instead of the wall, walk over and quietly drop a treat in between his paws. Chewing a toy is incompatible with chewing the wall: as long as he's chewing the toy, he's not chewing the wall.
If he's stealing cookies from the toddler, you first need to remove the opportunity so he's not rewarded. If the toddler is wandering around with a cookie and your dog has a food theft problem, try crating the dog for that brief period or gating him off in a different section of the house. That way, you're making sure he's not getting rewarded instead of just ignoring a self-rewarding behavior. The incompatible behavior here, by the way, would be "leave it." It is absolutely possible to teach a dog—without ever yelling at him or making his life unpleasant—that he needs to leave the toddler's cookie alone because there are other ways for him to get rewarded. A dog who knows "leave it" can learn to march right away from the toddler and over to you for food. Check out Susan Garrett's "It's Yer Choice" method for some ideas on teaching this kind of impulse control.
The next time you hear somebody complaining that positive trainers are permissive or don't have solutions for problems like jumping, biting, chewing, and stealing, remember that folks who say that either don't understand how nonrewarding works, or they've been watching ineffective trainers. When good trainers work on an undesired behavior, they're careful to identify any potential self-reward so it can be removed. And smart trainers look for the opportunity to teach the dog what they do want instead of just yelling at the dog for what they don't want.
To state the principle as simply as possible: nonreward the undesired behavior, and reward an incompatible, desired behavior. That should decrease the undesired behaviors as quickly as possible without intimidating or hurting the dog. And that's the goal, right?