Let $X$ be an integral model of a $\mathbb{Q}$-variety. If $X$ is proper, I found in a paper (page 1 in https://arxiv.org/pdf/2010.11763.pdf) that says $X(\mathbb{Z})=X(\mathbb{Q})$ from the valuation criterion of properness.
But from the criterion, all I can do is that the existence of a $\mathbb{Q}$-point in $X$ implies the existence of $\mathbb{Z}_{(p)}$-points for every prime number $p\in\mathbb{Z}$ where $\mathbb{Z}_{(p)}$ is the localization of $\mathbb{Z}$ by a prime ideal $(p)$.
My question is
(1) Does the existence of $\mathbb{Z}_{(p)}$-points for every prime number $p$ imply the existence of a $\mathbb{Z}$-point for such scheme $X$?
(2) Does the statement remain true when $X$ is a general proper scheme?
Any references of this fact or similar statement are welcome. I did not find one; maybe it is too easy to mention.