Just make sure this is what you mean: $P$ is the midpoint of chord $AB$, not arc $AB$ and the circle with $AB$ as diameter meets $C_1$ at more than one point, say $T_1$ and $T_2$, and both $PT_1$ and $PT_2$ are tangent to $C_1$. If so, the ratio of power of $P$ with regard to $C_1$ and $C_2$ is -1, and it is known that the locus of points whose powers w.r.t two fixed circles are in a constant ratio is a circle coaxal with them.

Let me put it this way: Let $ O_{1} $, $ O_{2} $ be the centers of $ C_{1} $ and $ C_{2} $, respectively, $ O $ the midpoint of $ O_{1} $$ O_{2} $, $ C_{1} $ and $ C_{2} $ meet at $ E $ and $ F $. The locus is portion of the circle with center $ O $ and radius $ OE $ or $ OF $.