Let $\pi\colon P\to X$ be some $G$-principal bundle; $\{U_\alpha\}$ a cover of $X$; and $\{s_\alpha\colon U_\alpha\to\pi^{-1}(U_\alpha)\}$ a collection of local sections.
Claim: The pullback $\pi^*P$ of $P$ over itself is trivial, as witnessed by the global section $\pi^* s_\alpha$.
My question: What does the above statement mean (if it makes sense at all)? That is, we know that the local sections differ by some transition $g_{\alpha\beta}\colon U_\alpha\beta\to G$, so how do we obtain a single global section from these local ones on the pullback?
The claim is correct, the local sections are irrelevant.
Proof of claim:
The pulled-back bundle $\pi^*(P) $ consists of those ordered pairs $(p,p')\in P\times P$ which satisfy $\pi(p)=\pi(p')\in P$.
The projection $\Pi:\pi^*(P)\to P $ of that bundle to its base is given by the formula $\Pi(p,p')=p.$
This projection has a canonical global section $\sigma:P\to \pi^*(P)$ given by $\sigma (p)=(p,p)$.
And thus the $G$-bundle $\pi^*(P)$ is trivial, like all principal bundles having a global section.