It is a famous consequence of Tsen's theorem that a smooth curve over an algebraically closed field has trivial Brauer group. But what about curves over non algebraically closed fields?
Let us fix a smooth, projective curve $X$ over some field $k$. If $X$ has a rational point $x\in X(k)$, then the natural map $\text{Br}(k)\to\text{Br}(X)$ has a retraction $\text{Br}(X)\to\text{Br}(k)$, thus it is injective. Moreover, it is widely known that $\text{Br}(X)$ injects in $\text{Br}(k(X))$. But what about closed points? A Brauer class that is trivial on each closed point is globally trivial?
Precise question: Is the map
$$\text{Br}(X)\to\prod_{x\in X^1}\text{Br}(k(x))$$
injective? If not, can we describe its kernel?
Observe that, for $k$ algebraically closed, the injectivity above is precisely $\text{Br}(X)=0$.
Best Answer
I don't think your map is injective. Here is an attempt at a recipe for constructing a counterexample.
The ingredients are a $C_1$-field $F$ of characteristic zero and a smooth projective curve $X_0$ over $F$ having non-trivial Brauer group. For a concrete example take $F=\mathbf{C}(t)$ and $X_0$ to be an elliptic curve with non-trivial Brauer group; a calculation of such a curve can be found in O. Wittenberg, Transcendental Brauer–Manin obstruction on a pencil of elliptic curves, PDF file here.
Edit: What's below is unnecessary. $X_0$ is already a counterexample, since every finite extension of $F$ has trivial Brauer group.
Given these ingredients, let $k$ be the field $F((t))$, let $\mathcal{X}$ be a smooth proper curve over $F[[t]]$ whose special fibre is $X_0$, and let $X$ be the generic fibre of $\mathcal{X}$.
There is a natural injective map $\mathrm{Br}(\mathcal{X}) \to \mathrm{Br}(X)$. Let $\alpha$ lie in the image of this map, and let $P$ be a closed point of $X$. If $R$ is the integral closure of $F[[t]]$ in $k(P)$, then $P$ extends to an $R$-point of $\mathcal{X}$, and $\alpha(P) \in \mathrm{Br}(k)$ lies in the image of $\mathrm{Br}(R)$, which is trivial ($R$ is a Henselian local ring whose residue field is $C_1$). Therefore $\alpha(P)=0$. This holds for all closed points $P$, so $\alpha$ lies in the kernel of your map.
It remains to show that $\mathrm{Br}(\mathcal{X})$ is non-trivial. For any $n$ coprime to the characteristic of $F$, proper base change gives $\mathrm{H}^2(\mathcal{X},\mu_n) \cong \mathrm{H}^2(X_0, \mu_n)$. Then the Kummer sequence shows that $\mathrm{Br}(\mathcal{X})[n] \to \mathrm{Br}(X_0)[n]$ is surjective. In particular, $\mathrm{Br}(\mathcal{X})$ is non-trivial whenever $\mathrm{Br}(X_0)$ is non-trivial.
Remark: if you don't insist that $X$ is a curve, then things are much easier: take a surface over a finite field having non-trivial Brauer group.