Linearly disjoint of local fields

Question

If $K$ and $L$ are two number fields such that $K\cap L = \mathbb{Q}$, then can we say something about the intersection of two local fields satisfy $K_{\mathfrak{B_1}}\cap L_{\mathfrak{B_2}} = \mathbb{Q}_p$, where $\mathfrak{B_1}\cap\mathbb{Q} = \mathfrak{B_2}\cap\mathbb{Q} = p $?
or "are there some conditions or theorems can make this fact hold?" I would appreciate some guidance.