Is semi-simplicity of Galois representations local?

Let $rho:G_{mathbb{Q}}rightarrow text{Gl}(V)$ be a finite dimensional $ell$-adic Galois representation. Then for each prime, by pre-composing $rho$ with the natural inclusion $G_{mathbb{Q}_p}rightarrow G_{mathbb{Q}}$, we get a representation of the local Galois group. I’m curious about how the property of being semisimple behaves under localization, that is to say if $rho$ is semisimple, can we conclude that $rhovert_{G_{mathbb{Q}_p}}$ is semisimple?

Conversely, if for all $pneq ell$ (or maybe all primes) $rho_{G_{mathbb{Q}_p}}$ is semisimple, does it follow that $rho$ is semisimple? If I’m not mistaken, all the $rho_{G_{mathbb{Q}_p}}$ being semisimple implies that $rhovert_{bigcup_p G_{mathbb{Q}_p}}$ is semistable, and since ${bigcup_p G_{mathbb{Q}_p}}$ is of finite index in $G_mathbb{Q}$ (I think), this would mean $rho$ is semisimple.