Let
${\mathbb{G}(n_1,n_2,m)}$ be a uniformly random m-edge subgraph of the complete bipartite graph
${K_{n_1,n_2}}$ with bipartition
$(V_1, V_2)$, where
$n_i = |V_i|$,
$i=1,2$. Given a real number
$p \in [0,1]$ such that
$d_1 \,{:\!=}\, pn_2$ and
$d_2 \,{:\!=}\, pn_1$ are integers, let
$\mathbb{R}(n_1,n_2,p)$ be a random subgraph of
${K_{n_1,n_2}}$ with every vertex
$v \in V_i$ of degree
$d_i$,
$i = 1, 2$. In this paper we determine sufficient conditions on
$n_1,n_2,p$ and m under which one can embed
${\mathbb{G}(n_1,n_2,m)}$ into
$\mathbb{R}(n_1,n_2,p)$ and vice versa with probability tending to 1. In particular, in the balanced case
$n_1=n_2$, we show that if
$p\gg\log n/n$ and
$1 - p \gg \left(\log n/n \right)^{1/4}$, then for some
$m\sim pn^2$, asymptotically almost surely one can embed
${\mathbb{G}(n_1,n_2,m)}$ into
$\mathbb{R}(n_1,n_2,p)$, while for
$p\gg\left(\log^{3} n/n\right)^{1/4}$ and
$1-p\gg\log n/n$ the opposite embedding holds. As an extension, we confirm the Kim–Vu Sandwich Conjecture for degrees growing faster than
$(n \log n)^{3/4}$.