I'm so glad you've asked this important question. Gratitude is an area that causes endless confusion in the wildly privileged life we lead in this country, and it becomes so easy to take our embarrassment of riches for granted. And so easy for kids to become overindulged.
Looking through your profile, it appears your kids are pretty young (2.5 – 4?). The behavior you describe is really completely normal in that age group, which is not to say that there's nothing you can do to improve it.
"Spoiled" is a pretty slippery concept, because it's basically a lack of appreciation. Gratitude is a feeling, like happiness, regret, anger, sadness or fear, and we can't simply insist that anybody feel, or not feel, any particular emotion. Even as adults, we don't have that kind of emotional control. But if we're thoughtful about it, we can help set up the conditions in which positive emotions are likely to arise.
The best way to teach gratitude is to model your own, and that can be for almost anything, like a nourishing meal, friendship, work, health, comfy shoes, soft rain in summer, a field full of bright dandelions. In fact, practicing thankfulness early and often for even small blessings is a wonderful way to experience happiness yourself, which in turn makes it more likely for those around you, especially your children.
We can punish kids for lack of gratitude, but that's like punishing them for lack of happiness. We can manipulate them emotionally when they "make" us feel bad, although that's actually a misconception – our own thoughts make us feel bad. But that won't make them feel anything more positive than guilt, and puts far too much power in their inexperienced hands, believing they can "make" us feel bad at will (which adults then interpret as spoiled or bratty).
We can get really restrictive about giving them treats, or we can take toys and belongings away from them when they don't demonstrate the feelings we deem appropriate, but that does little beyond weakening trust and communication, and confusing our kids. That's extremely unlikely to get the result of "Okay, Mom, I get it – I feel really, really appreciative now!"
So, there's a balance to look for as your kids get older. Besides modeling thoughfulness, kindness, respect and gratitude for and toward our kids, are there any other practices that would help? Respecting your own needs and reasonable boundaries is terribly important. I'd be strongly inclined to be clear – ahead of time – about what treats you will or won't buy your children on a shopping trip… IF they cooperate and don't keep begging.
If they beg and whine, remind them (you'll have to do this repeatedly while they're young) that you've already told them what treat they'll get, IF they cooperate. Then if they don't, tell them sympathetically that you understand that they want every good thing they see, but that they've lost their chance on this trip. If they cry and beg, after a warning or two, too bad. You'll have to repeat several times before they get it, especially for the younger one, and especially if begging and crying have worked for them in the past.
I don't know what you mean by others seeing the child's sorrow and giving in. Do "others" give your kids what they want, or do you, hoping to look like The Good Mom? If you're worried about what other people think, you're mind-reading a bunch of strangers to decide how to parent, and you'll never be consistent enough to let your kids know you mean what you say.
Children's behavior is largely about finding strategies to meet their perceived needs. So, instead of giving in, and instead of more intensive scolding, or even punishment from adults, kids generally respond positively and quickly to "I see by your behavior, you want/need more of X. I really, really understand. I wish I could give you as much X as you want. More, even! Now, I need to go and do Z. Help me get this done, and then we can enjoy some X."
My grandson's whining has miraculously disappeared since instituting this approach a couple of years ago (he's now 4.5). Though he does receive the logical consequences for his choices and mistakes, punishment as a parent-imposed ritual has disappeared from his life – there's no need for it. His parents (and other nurturers) don't have to force or manipulate him, or seldom even raise our voices, to get his calm, respectful, usually cheerful cooperation. He knows we're on his team, and so he's happy to be on ours.
If this sounds as wonderful to you as it is, teach yourself why children become stubborn and demanding starting around two, and how to respond, and you can melt away a tremendous amount of that youthful wheedling and resistance. Three fabulous resources for helping you in this follow:
The Happiest Toddler on the Block by Dr. Harvey Karp (you can also google this for some great little videos showing him putting his techniques into action);
Also the book Raising an Emotionally Intelligent Child by John Gottman, and the concept of Emotion Coaching, another term you can google for lots of useful information. (Here's one good link to get you started: http://www.education.com/reference/article/important-pare... .)
There are also some terrific books working with related techniques. One of my favorites is by Faber and Mazlish, How to Talk So Kids Will Listen, and Listen So Kids Will Talk. The methods make for good emotional connections, resulting in happy, cooperative and (mostly) obedient children. Every young family I know that has employed these methods have been pleased, and enjoy much more peaceful family life.