One sufficient condition is when $\Omega_1$ and $\Omega_2$ are bounded: Suppose $\Omega_1,\Omega_2\subset\Bbb{R}^m$ are such that $\overline{\Omega_1}$ and $\overline{\Omega_2}$ are disjoint, bounded subsets of $\Bbb{R}^m$. By the Urysohn lemma, there exists a continuous function $f:\Bbb{R}^m\rightarrow\Bbb{R}$ satisfying $f\restriction_{\overline{\Omega_1}}\,\equiv 1$ and $f\restriction_{\overline{\Omega_2}}\,\equiv -1$. Applying the Stone-Weierstrass theorem, there exists a polynomial $p$ such that $|p(x)-f(x)|<\frac{1}{2}$ for every $x\in \overline{\Omega_1}\cup\overline{\Omega_2}$. Thus $p(x)>\frac{1}{2}$ for $x\in\Omega_1$ while $p(x)<-\frac{1}{2}$ for $x\in\Omega_2$.

One sufficient condition is when $\Omega_1$ and $\Omega_2$ are bounded: Suppose $\Omega_1,\Omega_2\subset\Bbb{R}^m$ are such that $\overline{\Omega_1}$ and $\overline{\Omega_2}$ are disjoint, bounded subsets of $\Bbb{R}^m$. By the Urysohn lemma, there exists a continuous function $f:\Bbb{R}^m\rightarrow\Bbb{R}$ satisfying $f\restriction_{\overline{\Omega_1}}\,\equiv 1$ and $f\restriction_{\overline{\Omega_2}}\,\equiv -1$. Applying the Stone-Weierstrass theorem, there exists a polynomial $p$ such that $|p(x)-f(x)|<\frac{1}{2}$ for every $x\in \overline{\Omega_1}\cup\overline{\Omega_2}$. Thus $p(x)>\frac{1}{2}$ for $x\in\Omega_1$ while $p(x)<-\frac{1}{2}$ for $x\in\Omega_2$.

An immediate corollary: $\Omega_1,\Omega_2\subset\Bbb{R}^m$ can be separated by a polynomial if there exists a polynomial map $F:\Bbb{R}^m\rightarrow\Bbb{R}^k$ such that $F(\Omega_1),F(\Omega_2)$ are bounded subsets of $\Bbb{R}^k$ with disjoint closures.