We Buddhists like our emptinesses. But we have a problem with talking about emptiness in English, because "emptiness" is a noun. There is no such a thing as emptiness. It should be an adverb -- a word used to describe a quality.
Things are "empty" of three particular characteristics: things are not permanent; things are not separate from the causes and conditions in which they arise; and things are not ultimately satisfying. These are the three marks of all things.
We have this odd word “emptiness” that refers to very different things and experiences. Feeling empty -- feeling a lack of companionship -- is not the same as "all things are empty.” Though when I hear about how there is no thing that can make me ultimately happy, I feel a little lonely! But I think the loneliness is a different emptiness than the lack of permanence and satisfaction. When I’m lonely, or angry, there's something really vivid there!
Like everything that arises, feelings are empty (of permanence, separateness, and satisfaction), but feelings and things do exist -- they are experienced, they function. Tables are square and hard, eyes do see them, ears do hear birds that do fly in the sky, emotions do arise and feel good or bad.
The Heart Sutra says emptiness and form are not different. Things are not separate from their lack of certain characteristics. And things’ lack of those qualities is not separate from the things. You can't have a thing without its lack, and you can't have the lack without a thing.
Personally I find the language of the Heart Sutra to be a little one-sided. It keeps saying things don’t exist. They don’t exist the way we think they do, but that doesn’t mean they don’t exist. The English doesn’t really work well here. At least it’s the compassionate one Avalokiteshvara who’s telling me -- I keep trying to hear what he’s saying!
We humans also lack permanence and separate existence. This lack we call "non-self" -- another adverb masquerading as a noun! But the sense of self certainly does exist: the sensations and feelings and behaviors we call our self are experienced, they function, and they can cause all sorts of problems. When we are not grasping at our sense of self, "it" stops arising in ways that make us suffer. We are "selfless" even though we are still there. There is neither a self nor a non-self.
When we see that things are empty of the solidity that we usually try to impose, when we see the futility of trying to grasp and gain satisfaction from things that are impermanent, then we feel compassion. We have experienced the suffering that comes from grasping, and we wish that everyone could see the true nature of things, so that their confusion, grasping, and suffering would end.
This is all rather abstract. Yesterday I was arguing with a friend. Both of us were defending our sense of self, and experiencing hurt and anger. Then for a moment or two, I saw that I was defending an experience that lacked any permanence or any possibility of making me happy. For a moment I felt compassion -- here we were, stuck in our lousy sinking boats. And then my sense of self grabbed at its territory again, and confusion and anger rose right back up.
Experience is heart-breaking, language is tricky.