Let $G$ be a group acting transitively on sets $\Omega$ and $\Lambda$. Then there is a natural induced action of $G$ on cartesian product $\Omega \times \Lambda$. I can prove that if $G$ is finite this action is transitive if and only if $G = G_{\omega}G_{\lambda}$ for all $\omega\in\Omega, \lambda\in\Lambda$ (here $G_{\omega}$ and $G_{\lambda}$ are stabilisers of $\omega$ and $\lambda$, respectively).
My question. Is this true, when $G$ is infinite group?
Yes I believe so. This follows from two statenments.
If a group $G$ acts trantively on a set $X$, and $H \le G$, then $H$ acts transitively on $X$ if and only if $G = HG_x$ for all $x \in X$.
With your hypotheses, $G$ acts trasitively on $\Omega \times \Lambda$ if and only if $G_\omega$ acts transitively on $\Lambda$ for all (or equivalently some) $\omega \in \Omega$.
I don't really see how finiteness is involved in either of these claims. Which of these are you unsure of, and in which direction?