I never actually use Jeffreys priors really, generally using weakly informative priors instead. But one thing I've never quite understood about Jeffreys priors are where the Fisher information comes from.
For example, this page states that "construction is based on the Fisher information function of a model." which inclines me to think it's not based on the observed data, but on a set of plausible parameter values one might observe. But that contradicts the fact that it's an uninformative prior. On the other hand, I've never seen Jeffreys prior described as data dependent so using the Fisher information of the observed data doesn't quite make sense either.