= Solution
The two totals are independent, with $n\sim\operatorname{Bin}(2N,q)$ and $m\sim\operatorname{Bin}(2M,p)$. Their joint likelihood is
$$
L(p,q)=\binom{2N}nq^n(1-q)^{2N-n}
\binom{2M}mp^m(1-p)^{2M-m}.
$$
Differentiating the separated <log-likelihood> gives
$$
\boxed{\widehat p=\frac m{2M},\qquad\widehat q=\frac n{2N}.}
$$
These formulas include all-zero or all-one totals as boundary maxima. For interior $p,q$, the <Fisher information> in the ordered parameter vector $(p,q)$ is
$$
\mathcal I(p,q)=\begin{pmatrix}
\dfrac{2M}{p(1-p)}&0\\[2mm]
0&\dfrac{2N}{q(1-q)}
\end{pmatrix}.
$$
The off-diagonal entries vanish because the likelihood separates, and the two estimates are independent even at finite sample size. For large $M,N$, their joint distribution is approximately
$$
\boxed{\begin{pmatrix}\widehat p\\\widehat q\end{pmatrix}
\sim N\left(\begin{pmatrix}p\\q\end{pmatrix},
\begin{pmatrix}p(1-p)/(2M)&0\\0&q(1-q)/(2N)\end{pmatrix}\right).}
$$
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