Let $\xi$ and $\eta'$ be two independent identically Cauchy distributed random variables (Lebesque density $\frac 1\pi \frac{1}{1+x^2}$) and put $\eta=\xi$ (which are trivially not indenpendent). Then $\xi+\eta=2\xi$ and $\xi+\eta'$ have the same distribution which follows by Fourier transformation: The characteristic function of $\xi$ is $\varphi_\xi(t)=e^{-|t|}$ and thus $\varphi_{\xi+\eta}(t)=\varphi_{2\xi}(t)= \varphi_\xi(2t)= e^{-2|t|} =e^{-|t|}e^{-|t|}= \varphi_\xi(t) \varphi_{\eta'}(t)=\varphi_{\xi+\eta'}(t)$.

Let $\xi$ and $\eta'$ be two independent identically Cauchy distributed random variables (Lebesgue density $\frac 1\pi \frac{1}{1+x^2}$) and put $\eta=\xi$ (which are trivially not indenpendent). Then $\xi+\eta=2\xi$ and $\xi+\eta'$ have the same distribution which follows by Fourier transformation: The characteristic function of $\xi$ is $\varphi_\xi(t)=e^{-|t|}$ and thus $\varphi_{\xi+\eta}(t)=\varphi_{2\xi}(t)= \varphi_\xi(2t)= e^{-2|t|} =e^{-|t|}e^{-|t|}= \varphi_\xi(t) \varphi_{\eta'}(t)=\varphi_{\xi+\eta'}(t)$.