This is an extended comment:
As Misha says, the geodesics might be immersed in general, and in fact the pants and the surfaces they construct will in general be highly immersed (lots of self-intersections).
If you'd like to visualize embedded geodesics, you can imagine the lifts to the unit tangent bundle, or for a given pair of pants, it will lift to an embedded pants in some covering space corresponding to the image of the fundamental group $\rho(\pi_1(\Pi^0))$. To understand the feet of $\gamma_i$, then you can work in this covering space. The feet will be at the points of the shortest geodesics connecting the three boundary components in pairs (seams). There's a canonical involution sending $\gamma_i$ to its inverse, fixing the seams, and thus the feet lie equally spaced about each geodesic. Then project the whole picture back down into $M$ to get the feet.