Your question as originally written (which Henry correctly diagnosed as problematic in two ways) does not match the more reasonable aim reflected in your comments to Henry's answer. Specifically, your comments make it sound like you want to show that the satisfiability of both $\diamond X$ and $\diamond Y$ implies the satisfiability of $\diamond X \wedge \diamond Y$, **not** the satisfiability of $X\wedge Y$ as your original phrasing states.

If your notion of satisfiability of a formula $Z$ is simply that there is some Kripke model $\mathcal{M}$ (with no restrictions on its accessibility relation) and some world $w$ in it such that $\mathcal{M},w\models Z$, then this weaker form of the question isn't too difficult to answer.

Let $\mathcal{M},w\models\diamond X$ and $\mathcal{N},v\models\diamond Y$. In particular, there is a world $u$ in $\mathcal{N}$ which is accessible from $v$ and satisfies $Y$. Now just form a new model $\mathcal{P}$ whose set of worlds is the union of those of $\mathcal{M},\mathcal{N}$, and whose accessibility relation is the union of those of $\mathcal{M},\mathcal{N}$, plus we set $u$ to be accessible from $w$. Then $\mathcal{P},w\models\diamond X \wedge \diamond Y$.

Henry's point about underspecification is still pertinent. I'm not sure I've gotten at what you want, and if you were to be limited to special kinds of Kripke frames, for instance, then the argument would need to say a bit more (ensuring we end up with an appropriate $\mathcal{P}$). I hope this is helpful.