Foreword: 'Intentionality' means something quite different in philosopher-speak than it does in common English usage. 'Intentional' mental states are ones that are about something else. Beliefs, desires, etc., are all directed at things. That's what I mean by 'intentional' in this context. (But note that I use the word 'intend' in its usual sense.) Second-order intentional states are ones that are about (or directed at) some other intentional state. So if I have a belief about what you're thinking (about), then my belief is a second-order intentional state. Similarly, third-order intentional states are about second-order ones, and so forth. Now, I want to explain why genuine communication requires at least third-order intentionality.
Suppose my little brother intends for me to jump. He might (and sometimes does) achieve this by sneaking up behind me and yelling "Boo!". But that's not communication, in the fullest sense of the word. It would be quite a different sort of action were he to instead request of me, "please jump." (I don't think he'd find that nearly so fun, for one thing.) Such a speech-act would show not only that he intends me to jump, but also that he intends for me to recognize that he wants me to jump.
Purposive communication requires an intentional state of at least third-order complexity. The speaker wants his audience to recognize what the speaker intends by his utterance. Put another way, you don't just communicate 'X', you rather communicate, "I am trying to convey 'X'". (This is the difference between discreetly insulting someone, or making it clear to him that you want him to know you're insulting him.) Anything less would fail to qualify as 'communication', in the fullest sense of the word.
Note that genuine communication is thus sufficiently complex to allow for deception. Deception occurs when the utterer intends his audience to mistake what he truly intends. If my brother is choosing between two slices of cake, with the larger on the right, I might say "the one on the left sure looks good," intending for my brother to falsely believe that I want the left slice (thus tempting him to choose it himself).
Full deception, so defined, requires at least three orders (count and see). More simple trickery can be obtained with merely second-order intentions though, e.g. if I intend for you to believe something false about the world.
Where am I going with all this? Well, I think it's quite interesting in itself. But also, it could potentially help us to ascribe intentional states to animals -- e.g. if we observed deceptive behaviour in animals (of a sufficient complexity to rule out simple stimulus-response explanations).
One final note: second-order intentionality requires one to have a theory of mind. You can't really intend to implant false beliefs unless you have the concept of 'belief' to begin with. And it's pretty controversial whether any non-human animals have a theory of mind. So skeptics are going to be even more reluctant to ascribe full-blown communicative or deceptive intentions to animals. Ah well, interesting stuff nonetheless.
Update: a further thought: it seems like there's something a bit more humble about the higher-order intentions of genuine communication. Suppose I want to inform you that it's raining outside. A second-order description might say that I intend for you to believe that it's raining outside. But I could achieve this through coercively brainwashing you, for example. Genuine communication doesn't allow for such brainwashing -- that's not what it's about. When I say to you: "It's raining outside," I rather intend for you to recognize that I believe it's raining outside, or something like that. It's more indirect, and it's up to you to do with the information what you will. Once you've recognized my communicative purposes, my job is done -- I don't have any further desire to force you to accept the information. Brainwashing you wouldn't achieve my goals here at all.