Does $\,x>0\,$ hint that $\,x\in\mathbb R\,$?

It really depends on context. But be safe; just say $x > 0, x\in \mathbb R$.

Omitting the clarification can lead to misunderstanding it. Including the clarification takes up less than a centimeter of space. Benefits of clarifying the domain greatly outweigh the consequences of omitting the clarification.

Besides one might want to know about rationals greater than $0$, or integers greater than $0$, and we would like to use $x \gt 0$ in those contexts, as well.

ADDED NOTE: That doesn't mean that after having clarified the context, and/or defined the domain, you should still use the qualification "$x\in \mathbb R$" every time you subsequently write $x \gt 0$, in a proof, for example. But if there's any question in your mind about whether or not to include it, error on the side of inclusion.

One might be able to decode that notation, but why force this puzzle on the reader?

Easiest solution is to just say $$ x\in\mathbb{R}^+ $$ Expresses both conditions in one hit.

There are ordered fields which strictly extend the real numbers, there $x>0$ is meaningful, but need not imply $x\in\Bbb R$.