Let $O_1,O_2$ be the centers of $\mathcal{C}_1,\mathcal{C}_2.$ Let $M,N$ be the midpoints of $\overline{P_1P_2},\overline{A_1A_2}$ and $X_1,X_2$ the midpoints of $\overline{A_1P_1},\overline{A_2P_2},$ i.e. the projections of $O_1,O_2$ on $A_1P_1,A_2P_2,$ respectively. Let $K$ be the midpoint of $\overline{MN}$ and the perpendicular bisector $\ell$ of $\overline{MN}$ cuts $A_1P_2,A_2P_2$ at $Y_1,Y_2,$ respectively. Since $MN$ is midparallel of $A_1P_1 \parallel A_2P_2,$ then $K$ is midpoint of $\overline{X_1X_2}$ and $\overline{Y_1Y_2}$ $\Longrightarrow$ $X_1Y_1=X_2Y_2,$ which means that $O_1$ and $O_2$ are equidistant from $\ell$ $\Longrightarrow$ $\ell$ goes through the midpoint $O$ of $\overline{O_1O_2}$ $\Longrightarrow$ locus of $M$ is the circle with center $O$ and radius $ON.$