When can real-valued random variables $X$ and $Y$ be expressed as $X' + Z$ and $Y' + Z,$ where $Z$ is non-degenerate and independent of $X'$ and $Y'$?

Question

...and, are there any interesting results re how much of the variance/dispersion of $X$ and $Y$ can be loaded onto a shared, independent $Z$ in this way? E.g. relating it to the covariance of $X$ and $Y?$

I'm particularly interested in situations where $X$ and $Y$ are heavy-tailed, but can be expressed as $X' + Z$ and $Y' + Z$ where $Z$ (independent of $X'$ and $Y'$) is heavy-tailed, but $X'$ and $Y'$ aren't (and what properties of $X$ and $Y$ might tell you that you're in such a situation).

I know this is a bit open-ended; happy to be directed to textbooks or other resources that discuss this sort of thing, in lieu of particular results. Thanks!