  We describe a general correspondence between weighted minimal surfaces in
$\mathbb{R}^3$ and weighted maximal surfaces with some admissible singularities
in $\mathbb{L}^3$, for a class of functions $\varphi$ which provides the
corresponding weight. For these families of surfaces, we provide a Weierstrass
representation when $\dot{\varphi}\neq 0$ and analyze in detail the asymptotic
behavior of both such a weighted maximal surface around its singular set and
its corresponding weighted minimal immersion around the nodal set of its angle
function, establishing criteria that allow us to easily determine the type of
singularity and classify the associated moduli spaces.
