Dogs often very clearly show that their food belongs only to them. It is a natural instinct. It is evolutionary heritage from wolves. But since you must be the leader, food belongs to you. It does not mean that you will not give it to your dog, but, if you wish to take it from it, it must let you without any resistance. It is important because of its upbringing, but also to avoid danger of your dog eating something it should not – especially outside.
It may happen that they will hide the food before eating it, carrying it to another room or burring it in the yard. One thing is to hide leftovers that in certain moment they cannot eat because they do it to protect imaginary stockpile, but such behaviour can develop into possessiveness.
In essence, possessiveness is a need to obtain something, when the dog is not motivated by objective reason such as hunger, danger, female, mating...
Except food, some dogs pile toys or other objects hiding them in corners or simply displaying them in the centre of the room proving that they own them... Do not allow such behaviour since the behavioural problem that follows is – aggressiveness.
Possessiveness might be related to the owner too. There is a difference – sometimes barely noticeable – between guarding the owner and being possessive of the owner, which by itself is a problem, and besides that, shows that you are not the leader.