Dan didn't write a paper, he just made a comment in the discussion paper of Beskos et al. I guess I didn't really make that clear (now edited).

Did you mean $dZ_t = \hat{\mu}(Z_t)dt + \hat{\sigma} dW_t$, where $\hat{\sigma}$ is constant?

Thanks, this is the type of idea I was looking for. Yes, I can see that this kind of strategy will not work for many SDEs. Perhaps one could construct a counterexample based on Tanaka's formula. I'll take a look at the Lamperti transform.

Hm, Girsanov's theorem isn't really what I'm looking for. It tells you how to transform the laws of $X$ and $W$, whereas I'm looking for a transform on the paths themselves, if such a thing exists. That said, it wasn't me who voted your answer down.

Darryl Holm's two-volume "Geometric Mechanics" may be worth looking at.