No, it is possible for $X$ not to have points everywhere locally and still have every $k$-rational divisor class be represented by a $k$-rational divisor (this is the "Picard equality" you want).

For instance, the Picard equality holds if $X$ has points everywhere locally except at a single place. Many examples of genus one curves satisfying this property -- over any number field $k$ -- are given in this paper of mine.

There should be no lack of other examples as well. In some sense the simplest family is as follows: it is enough to find hyperelliptic quartic curves $C: y^2 = P_4(x)$ which do not have points everywhere locally and for which the Jacobian elliptic curve $E$ has $E(k)$ finite of odd order. The point here is that then the "elliptic Kummer sequence"

$0 \rightarrow E(k)/2E(k) \rightarrow H^1(k,E[2]) \rightarrow H^1(k,E)[2] \rightarrow 0$

gives an isomorphism $H^1(k,E[2]) \rightarrow H^1(k,E)[2]$. This means that for each torsor $C \in H^1(k,E)[2]$ there is a unique rational divisor class of degree $2$, so $C$ comes from a hyperelliptic quartic curve iff this divisor class is represented by a rational divisor.

Actually, a simpler argument can be given if we assume a little more, that $E(k) = \{0\}$. (By a recent theorem of Mazur and Rubin, such curves exist over every number field $k$.) Then the degree zero part of $H^0(\operatorname{Pic} C) = E(k) = 0$, so the equality of $H^0(\operatorname{Pic} C) = \operatorname{Pic}(C)$ for torsors is equivalent to having period equals index, so if you can find a hyperelliptic quartic curve $C$ with Jacobian having trivial Mordell-Weil group and failing to have points over more than one completion, this will give an example of what you want.

In practice, it should be easy to write down such curves $C$ over $\mathbb{Q}$, say.

Note that the situation is different for genus one curves over a $p$-adic field $k$: by a theorem of Roquette-Lichtenbaum, the order of $C$ in $H^1(k,\operatorname{Jac} C)$ is equal to the order of $H^0(\operatorname{Pic} C)/ \operatorname{Pic}(C)$.