Solution:

Label the vertices of the hexagon $a b c d e f$.
The numbers that are larger than both of their neighbors can't be adjacent, so assume (by rotation) that these numbers take up slots $a c e$. We also have that $6$ and $5$ cannot be smaller than both of their neighbors, so assume (by rotation and reflection) that $a=6$ and $c=5$.
Now, we need to insert $1,2,3,4$ into $b, d, e, f$ such that $e$ is the largest among $d, e, f$. There are $4$ ways to choose $b$, which uniquely determines $e$, and $2$ ways to choose the ordering of $d$, $f$, giving $4 \cdot 2=8$ total ways.