  In this paper we introduce and study a bilinear spherical maximal function of
product type in the spirit of bilinear Calder\'{o}n-Zygmund theory. This
operator is different from the bilinear spherical maximal function considered
by Geba et al. We deal with lacunary and full versions of this operator, and we
prove weighted estimates with respect to bilinear weights. Our approach
involves sparse forms following ideas by Lacey, but we also use other
techniques to handle the particular triplet $(2,2,1)$.
