Let P be the property of "being an integral domain". Then
1: If $A$ is an integral domain, then $A_p$ is an integral domain for every prime ideal $p\subseteq A$.
On the other hand.
2: Let $A=A_1\oplus A_2$ be a direct sum of two integral domains. Then it is obviously not an integral domain, although $A_p$ is an integral domain for every prime ideal $p\subseteq A$. So P is not local.