Bryan Caplan has a post up on ideological Turing tests. Long story short, back in the 70s hawks evaluated doves as furnishing exaggerated positions of their views more often than doves said that about hawks.
The most obvious thing that could be behind this is that doves have less realistic views of their ideological opponents. But obvious isn't always right (and there doesn't seem to be any particular reason for this to be right - which should make us more suspicious) so let's look a little deeper.
I don't know what the explanation is but I like all of these better than the raw answer that doves exaggerate more than hawks, because none of them require me to assume the sincerity, open-mindedness, or intelligence are non-randomly distributed across the ideological spectrum*.
1. Selection problems
Another option is that the set of people self-identifying as hawks or doves is different from the set of people identified by out-group members as hawks or doves. The out-group may be very accurately describing a somewhat different group.
For example, a lot more people consider themselves "libertarian" than regular readers of EconLog would probably consider libertarian. As a non-libertarian I typically think of the sort of people who agree with what's on EconLog as "libertarian" and would write my summary of libertarian views using that mental definition. And it's a fine mental definition, I'd argue - one that many libertarians use to characterize themselves. But if Greg Mankiw were reading it (he has referred to himself as a libertarian) it wouldn't describe him very well and he'd say it's extreme.
Same with Glenn Beck. Same with Rand Paul. I have a facebook friend I've known since high school (i.e. - not picked up as a friend in my years of blogging) who calls himself a libertarian but isn't exactly in line with the community of libertarians I'm often butting heads with. If any of these people were reading my summary they'd think I was presenting an extreme position even if I described the EconLog libertarian position quite well.
Ideology is not a characteristic like height or weight that can be precisely measured or assigned.
Classifying people based on a survey rather than using self-identification doesn't really help either because even though you clean out the self-identification you still aren't cleaning out what I call a "libertarian".
This is also relevant to "liberal" which is probably more of a bad word in America than "conservative" (where I live neither seems to have any more stigma than the other but I live in a funny place - I am told this is the case in most of the country). People who conservatives would identify as "liberals" probably think of themselves more as "moderate". In that sense the people who self-identify as "liberal" probably are more outspoken than the average person in the group that a conservative would call "liberal".
There may also be signaling with buzz words and the structuring of the language. It may be that hawks respond more to buzz words and the use of certain language than doves are so they are more clued in when someone tries to imitate that. I'm on record saying that Lind/Dionne pose important questions but even I wince at how they write it up because I know they simply don't speak the same language. If one groups has more buzz-word-based discourse than another they may easily identify out-group members and complain that they are communicating an extreme position.
3. Garbage in, garbage out
Before assuming anything about how doves treat hawks we also have to consider the information that doves are basing their assessments on. It may be that hawks actually have fairly moderate and sophisticated views but in the public sphere, because of its rhetorical power, they communicate with extreme arguments. A dove assuming this is sincere might make an entirely faithful rendition of that argument that the hawk (who is now not making a public appeal) sees as too extreme. Maybe we can blame the hawk here. Maybe we can blame the dove for being credulous or not digging deeper. But we can't say that the dove is misrepresenting the hawk position as it has been communicated.
* We can think of extreme exceptions to this. Almost no one in the U.S. is a Marxist anymore except for people who are intellectually immersed in the legitimately important works of Karl Marx... which means that professors are much more likely to be Marxist. Intelligent people select into higher education so maybe Marxists are more intelligent on average than other people. But that's a very narrow explanation and shouldn't generally apply - and anyway it's not causal from intelligence to Marxism it's just an artifact of intellectual history.