As a kid, at home we knew what snacks we were allowed (two cookies after getting home from school, for example) but we were not allowed to randomly snack on stuff, and we were specifically not allowed to eat the hour before a meal. Water and milk were available for drinking, but sugary drinks required permission, and pop was not a weekday drink. I remember asking and getting permission quite frequently - a popsicle on a summer afternoon, a snack at night if I was still hungry, so we certainly weren't being deprived. The food wasn't hidden or locked away.
As a guest, we would be allowed to help ourselves from offered food (a plate of cookies passed around, etc), within reason, but not beyond that. Getting a drink from the fridge, or looking for a snack that wasn't out on offer would have been absolutely forbidden, and I remember my parents' less than complimentary comments when the child of guests tried rooting around in the fridge for something.
I can see why parents would want to control/monitor their kids' snacking. My nephew is five, and if he were given free range of the snack cupboard, he would eat nothing but junk food. Even with healthy snacks, he'd randomly eat snacks and then have no interest in meal times.