Well this is a very interesting question. Here is my two cents:

Let \( p_{ab} = \Pr\{A = a, B = b\}, a,b \in \{0, 1\} \)

Usually we will collect the frequency data and put in a two by two table like this:

\( \begin{tabular}{|c|c|c|}

\hline & $A = 0$ & $A = 1$ \\ \hline

$B = 0$ & $W$ & $Y$ \\ \hline

$B = 1$ & $X$ & $Z$ \\ \hline

\end{tabular} \)

and we know that the counts \( (W, X, Y, Z) \sim \text{Multinomial}(n;p_{00}, p_{01}, p_{10}, p_{11}) \)

So the MLE is \( (\hat{p}_{00}, \hat{p}_{01}, \hat{p}_{10}, \hat{p}_{11})

= \left(\frac {W} {n}, \frac {X} {n}, \frac {Y} {n}, \frac {Z} {n}\right) \)

Note that the parameters satisfy \( p_{00} + p_{01} + p_{10} + p_{11} = 1 \) and the counts satisfy \( W + X + Y + Z = n \). They both have 3 degrees of freedom.

Now when you only observe the value of the marginal counts \( (W + X, W + Y) \) (and the sample size \( n \) of course), the dimension of the statistic is less than the number of parameters now and surely you have loss some information. I think it is somehow like a censored data/missing data which you would see below.

To find the MLE in this case, first I try to write the likelihood:

\( L(p_{00}, p_{01}, p_{10}; a, b) = \Pr\{W + X = a, W + Y = b\} \)

\( = \sum_{w = 0}^{\min\{a, b\}} \Pr\{W = w, X = a - w, Y = b - w\} \)

\( = \sum_{w = 0}^{\min\{a, b\}} \frac {n!} {w!(a-w)!(b-w)!(n-a-b+w)!}

p_{00}^w p_{01}^{a-w} p_{10}^{b-w} (1 - p_{00} - p_{01} - p_{10})^{n - a - b + w} \)

Then some how you have a likelihood here and you may try to find the MLE by (numerically) maximizing the function. I am not sure if this function is easy to maximize numerically as it is a high degree polynomial. To combine the information from different independent trials we just multiply the likelihood together as before.

If \( n = 10, a = 3, b = 2 \), the above likelihood should be like the objective function I posted here:

http://www.talkstats.com/showthread.php/43739-Numerical-Multivariate-Maximization
For other references you may also look at EM algorithm for missing data and the MLE of survival curves of censored data.