Data blurring: sample splitting a single sample

Suppose we observe a random vector $X$ from some distribution $P$ in a known family with unknown parameters. We ask the following question: when is it possible to split $X$ into two parts $f(X)$ and $g(X)$ such that neither part is sufficient to reconstruct $X$ by itself, but both together can recover $X$ fully, and the joint distribution of $(f(X),g(X))$ is tractable? As one example, if $X=(X_1,\dots,X_n)$ and $P$ is a product distribution, then for any $m