A system of semilinear parabolic stochastic partial differential equations with additive space‐time noise is considered on the union of thin bounded tubular domains $D_{1,\eps}:=\Gamma\times(0,\eps)$ and $D_{2,\eps}:=\Gamma\times(-\eps,0)$ joined at the common base $\Gamma \subset {\R}^{d}$, where $d\ge1$. The equations are coupled by an interface condition on $\Gamma$ which involves a reaction intensity $k(x',\eps)$, where $x = (x',x_{d+1}) \in \mathbb{R}^{d+1}$ with $x' \in \Gamma$ and $|x_{d+1}| < \eps$. Random influences are included through additive space‐time Brownian motion, which depend only on the base spatial variable $x' \in \Gamma$ and not on the spatial variable $x_{d+1}$ in the thin direction. Moreover, the noise is the same in both layers $D_{1,\eps}$ and $D_{2,\eps}$. Limiting properties of the global random attractor are established as the thinness parameter of the domain $\eps \to 0$, i.e., as the initial domain becomes thinner, when the intensity function possesses the property $\lim_{\eps\to0}\eps^{-1}k(x',\eps)=+\infty$. In particular, the limiting dynamics is described by a single stochastic parabolic equation with the averaged diffusion coefficient and a nonlinearity term, which essentially indicates synchronization of the dynamics on both sides of the common base $\Gamma$. Moreover, in the case of nondegenerate noise we obtain stronger synchronization phenomena in comparison with analogous results in the deterministic case previously investigated by Chueshov and Rekalo [EQUADIFF‐2003, F. Dumortier et al., eds., World Scientific, Hackensack, NJ, 2005, pp. 645–650; Sb. Math., 195 (2004), pp. 103–128].