My answer is from the point of view of an UK academic statistician. In particular, as an academic that gets judged on advances in statistical methodology.
What would make me (or any other
scientist) a better collaborator?
To be blunt - money. My time isn't free and I (as an academic) don't get employed to carry out standard statistical analysis. Even being first/last author on a paper that uses standard methodology is worth very little to me (in terms promotion and my personal research). Paying for my time will buy me out of administrative or teaching duties. Payment could be through a joint grant.
In the UK, every five or so years academics have to submit their four best papers. My papers are judged on their contribution to the statistical literature. It sucks, but that's the way it is.
Now it may well be that you have a very interesting problem which would lead to advances in statistical techniques. However, just think about the size of your statistics department compared to the rest of the Uni. There probably won't be enough statisticians to go around.
In saying that, I do try and do some "statistical consultancy" once a year to broaden my interests and to help for teaching purposes. This year I did some survival analysis. However, I've never advertised this fact and I still get half dozen requests each year for help!
Sorry for being so negative :(
Specifically, what is one statistics
concept you wish all of your scientist
collaborators already understood?
That statisticians do statistical research. As one of my collaborators said:
Surely there's nothing left to solve in statistics?