That the random matrices $X_1X_1'$ and $X_2X_2'$ are iid is a special case of the following general facts: 

>**Fact 1:** Suppose that $X_1$ and $X_2$ are independent random variables (r.v.'s) defined on a probability space $(\Omega,\mathcal F,\mathbb P)$ with values in measurable spaces $(S_1,\mathcal S_1)$ and $(S_2,\mathcal S_2)$. For each $i\in\{1,2\}$, let $g_i$ be a measurable map from $(S_i,\mathcal S_i)$ to a measurable space $(T_i,\mathcal T_i)$. Then the r.v.'s $g_1\circ X_1$ and $g_2\circ X_2$, with values in the measurable spaces $(T_1,\mathcal T_1)$ and $(T_2,\mathcal T_2)$, are independent. 

>**Fact 2:** Suppose that $X_1$ and $X_2$ are identically distributed r.v.'s defined on a probability space $(\Omega,\mathcal F,\mathbb P)$ with values in a measurable space $(S,\mathcal S)$. Let $g$ be a measurable map from $(S,\mathcal S)$ to a measurable space $(T,\mathcal T)$. Then the r.v.'s $g\circ X_1$ and $g\circ X_2$, with values in the measurable space $(T,\mathcal T)$, are identically distributed.