Let $k$ be an algebraically closed field of characteristic $p\geq 0$ and $\ell$ a prime different from $p$. For a connected scheme of finite type over $k$ with geometric point $x$, and a lisse $\overline{\mathbb{Q}}_{\ell}$-sheaf $F$ on $X$ one can compute the global sections as follows $$H^0(X,F)=F_x^{\pi_1(X,x)}$$ I don't know where a proof of this is written down, but it seems to me that it works like this: Consider $F$ as projective system $(F_n)$ of $\mathcal{O}_E/\mathbf{m}^n$-sheaves, for $E$ a finite extension of $\mathbb{Q}_{\ell}$, and $\mathbf{m}$ the maximal ideal of $\mathcal{O}_E$. Each $F_n$ corresponds to a étale covering $X_n$ of $X$, and $(F_n)_x$ is the set of geometric points of $X_n$ over $x$. Now the sections of $X_n\rightarrow X$ correspond to points in $(F_n)_x$ which are fixed by the $\pi_1(X,x)$-action. Passing to the limit proves the formula for $H^0$.

My question is: Does this also hold for $k$ not algebraically closed?

I've never seen it stated like this, but my argument doesn't seem to use the fact that $k=\bar{k}$.

I think it is true for smooth curves, because a smooth curve $X\neq \mathbb{P}^1$ over any field of characteristic $p\geq 0$ is even an "étale $K(\pi_1(X,x),1)$ space", i.e. $$H^n(X,\mathbb{Z}_{\ell})=H^n(\pi_1(X,x), \mathbb{Z}_\ell)$$ for all $n\geq 0$.

finiteetale cover. With this assumption, your argument is correct (without any hypotheses at all on $X$, I should imagine), essentially because you're considering lisse l-adic sheaves over the finite etale site, and the latter is equivalent to the site of finite $\pi_1(X,x)$-sets. – Keerthi Madapusi Pera Oct 11 '11 at 14:07`$X\times_k\overline{k}$`

, and Deligne doesn't usually write what he doesn't need. – Alex Oct 12 '11 at 13:29