Let $M$ be a compact 3-manifold. I am interested in necessary/sufficient conditions for $M$ to have the following property: If $M = M_1 \cup_S M_2$ where $S$ is a connected surface and $M_1,M_2$ are connected compact 3-dimensional submanifolds, then $\pi_1(M_1)$ and $\pi_1(M_2)$ do not contain any 2-torsion.
I know that this is true when $M = S^3$ but I do not know how to prove this in other cases. I might guess that the above property is equivalent to $\pi_1(M)$ not containing any 2-torsion. One sufficient condition is that $\pi_2(M_1) = \pi_2(M_2) = 0$ for any such decomposition - but I want a condition that does not look at all possible splittings.