let $X_{1}$, $X_{2}$, $X_3$, $X_{4}$ be iid bernoulli rvs with $\mathbb{P}(0)=0.5$, $\mathbb{P}(1)=0.5$.
$Y_{1} = X_{1}+X_{2}+X_{3}$ and $Y_{2}=X_{1}+X_{2}+X_{4}$
$Y_{1}$, $Y_{2}$ are dependent binomial rvs by definition. I need to find max likelihood estimate of $X_{1}$ given I have observed $Y_{1}$ and $Y_{2}$.
how should I go about it?
There are two different cases.
If the observed values differ by $1$, this implies known values for $X_3$ and $X_4$, and the lower of the two observed values is the value of $X_1+X_2$. If this is $0$ or $2$, it implies the value $0$ or $1$, respectively, for $X_1$; if it is $1$, both values of $X_1$ are equally likely.
On the other hand, if the observed values are equal, this implies that the values of $X_3$ and $X_4$ are equal. Given that they are equal, the problem effectively becomes finding the maximum likelihood estimate for $X_1$ given $X_1+X_2+X_3$. If the observed value is $0$ or $3$, that determines $X_1$ with value $0$ or $1$, respectively. If the observed value is $1$ or $2$, then the value of $X_1$ is $0$ or $1$, respectively, with probability $2/3$, and $1$ or $0$, respectively, with probability $1/3$, so the maximum likelihood estimate in this case is $0$ or $1$, respectively.
In summary, the maximum likelihood estimate for $X_1$ given the two observed values is given by
$$ \begin{array}{c|cccc} &0&1&2&3\\\hline 0&0&0&-&-\\ 1&0&0&=&-\\ 2&-&=&1&1&\\ 3&-&-&1&1 \end{array} $$
where $-$ indicates a case that cannot occur and $=$ indicates that both values of $X_1$ are equally likely.