Number of LWE samples in NewHope
They're actually sampling $5n$ elements from $\Psi_{16}$. Perhaps Protocol 2 on page 5 shows this most clearly, where $\textbf{s}, \textbf{e} \stackrel{\$}{\leftarrow} \Psi_{16}^n$ and $\textbf{s}', \textbf{e}', \textbf{e}'' \stackrel{\$}{\leftarrow} \Psi_{16}^n$ are sampled (on line 3 on Alice' side, and line 1 on Bob's).

This probably also answers part of your follow-up question, but for completeness: the values $\textbf{s}$ and $\textbf{s}'$ are sampled independently by Alice and Bob, and are thus not at all related. The final shared secret is neither $\textbf{s}$ nor $\textbf{s}'$, but the reconciled $v$.