Remark

Clearly, we have $P(K, L) \subset P(K \cup L, K \cup L)$; therefore for $G$ to operate properly on a locally compact space $X$, it is sufficient that, for each compact subset $K$ of $X$, the set $P(K, K)$ is relatively compact in $G$. In particular, a discrete group $G$ operates properly on a locally compact space $X$ if and only if, for each compact subset $K$ of $X$, the set of $s \in G$ such that $s.K \cap K \neq \emptyset$ is finite.