Suppose you have two simple Ar[1] series of the form $y_n=y_{n-1}+e_n$ and $x_n=x_{n-1}+m_n$, where $e_n$ and $m_n$ are normal white noise processes with no auto-correlation and $Corr(e_n,m_n)=p$. Then suppose we have possibly non-overlapping data for Y and X (IE, observation 10 exists for Y but not for X), and to avoid data generating process issues, assume that the distribution of missing data is random.

Is there any way to estimate p?

As a follow-up question, is there a way to easily generalize to a situation where $y_n$ and $x_n$ are observed with known normally distributed measurement error?