If you want a representation $V$ to be self-dual up to a character, then either $S^2V$ or $\Lambda^2V$ (considered as representations of $G$) should have a 1-dimensionanal summand (corresponding to the isomorphism $V \to V^*\otimes\chi$). But as it was mentioned by Jim there are a lot of representations $V$ for which both $S^2V$ and $\Lambda^2V$ are irreducible. For example, this is the case for $G = SL(n)$ and $V$ the standard representation --- the example suggested by Bruce.

If you want a representation $V$ to be self-dual up to a character, then either $S^2V$ or $\Lambda^2V$ (considered as representations of $G$) should have a 1-dimensional summand (corresponding to the isomorphism $V \to V^*\otimes\chi$). But as it was mentioned by Jim there are a lot of representations $V$ for which both $S^2V$ and $\Lambda^2V$ are irreducible. For example, this is the case for $G = \operatorname{SL}(n)$ and $V$ the standard representation — the example suggested by Bruce.