  In this paper we investigate the intersection problem for $1$-surfaces
immersed in a complete Riemannian three-manifold $P$ with Ricci curvature
bounded from below by $-2$. We first prove a Frankel's type theorem for
$1$-surfaces with bounded curvature immersed in $P$ when $\text{\rm Ric}_{P} >
-2$. In this setting we also give a criterion for deciding whether a complete
$1$-surface is proper. A splitting result is established when the distance
between the $1$-surfaces is realized, even if $\text{\rm Ric}_{P} \geq -2$. In
the hyperbolic space $\mathbb{H}^3$ we show strong half-space theorems for the
classes of complete $1$-surfaces with bounded curvature, parabolic
$1$-surfaces, and stochastically complete $H$-surfaces with $H<1$. As a
by-product of our techniques a Maximum Principle at Infinity is given for
$1$-surfaces in $\mathbb{H}^3.$
