I would like to see an example of a sigma-compact Polish group which is not locally compact.
I know that e.g. $l^{\infty}$ is a topological group which is sigma-compact but not locally compact. But this is not a Polish group since it is not a $G_{\delta}$ subset of the Polish group $\mathcal R^{\omega}$.
It seems the following.
Each $\sigma$-compact Polish group $G$ is locally compact. Indeed, let $G=\bigcup K_n$ is a union of its compact subsets $K_n$. Since $G$ is a Polish space, it is Baire, so there exists an index $n$ such that the set $K_n$ has non-empty interior $K_n^\circ$. Pick an arbitrary point $x_0\in K_n^\circ$. If $x\in G$ is an arbitrary point, then $xx_0^{-1}K_n$ is a compact neighborhood of the point $x$.