The first sentence of the Wikipedia entry says: "Cognitive dissonance is an uncomfortable feeling caused by holding conflicting ideas simultaneously." Strictly speaking, holding and even entertaining them is not a problem unless you actually believe in the ideas to some extent. After all, you don't have to have anything invested in an idea just to think about it, and you can understand that if someone believed this or that then such and such would follow without having it be your belief. But as CJ points to, it really ends up being cognitive dissonance for both parties if it's some tug of war thing. The process is not just in one brain, but it becomes a negotiative process between the individuals involved, both trying to come to some balance when none of it needs to be identified with in the first place.
This is why it's funny to see people repeatedly make the obligatory "I agree," "I disagree" or "everyone is entitled to their belief" comments, where they don't actually looking at what the ideas say. The first two is only a statement of position, and often we see those responses (which might start with such a proclaimation) say pretty much nothing about the original message, and likewise the third is a kind of conversation stopper. I say if all people wants to do is to put their ideas out there to the world, repeatedly, over and over... sure, go right ahead - but if you're not actually interested in discussing anything it might as well be a kind of natural automatic disfellowshipping right there. Why should anyone acknowledge you in any way when you are really not interested in acknowledging what they have to say in any meaningful way? Oh yes, we can also continually repeat "well that's your opinion and I have mine" but what is the point of even saying it after a while? Just to project some sense of how civil you are? The natural outcome of this would seem to be that one starts to resemble an automated process, little more than a chat bot if you will.
Or to put it another way, if the unexamined life is not worth living, as Socrates said, then is the unexamined mind worth talking to?