combining dependent likelihoods

Question

Let $\vec{X}_{1,2}$ be random variable vectors. Suppose I have two data sets, named $\Omega_{1,2}$ respectively. I define the likelihoods for these two data sets to be $L_{1,2}(\vec{\theta}_{1,2}|\vec{x}_{1,2})$ for some model parameters $\vec{\theta}_{1,2}$. Suppose that the datasets overlap i.e., $\Omega_{1}\cap\Omega_{2} \neq \emptyset$ and the random variable vectors may contain an overlap in their components (i.e., $\exists i,j \in \mathbb{N} : (X_{1})_{i} = (X_{2})_{j}$). Similarly the model parameters may not coincide with one another, but may have overlap. Is there a rigorous prescription to combine the likelihoods for these dependent scenarios, and if so, can anybody point to literature on the subject? Thanks!