Here, we are talking about conservation of momentum for 2 rigid/elastic spherical bodies.
Conservation of momentum means that if there is no net external force acting on the system then the total momentum of the system remains constant.
In the case of rigid spherical bodies, the spheres come in contact with each other only at one point. This is because rigid bodies do not deform and two spheres (or even circles) can touch each other only at one point. Hence there is no question of how they touch each other.
But that was the case for perfectly rigid bodies, the case is not exactly same for perfectly elastic bodies. The important property of perfectly elastic bodies is that, they do not store/absorb any energy during deformation when they come in contact with each other. hence effectively, they can be treated as rigid bodies as they do not deform, hence don't store/absorb energy due to deformation.
Alternatively, perfectly elastic bodies can be imagined as two rigid bodies, one of which connected by a spring (figure)
The important thing to understand here is that if there is no external force acting on the system, Total Momentum Of The System Is Conserved. But that is not always the case with Total Energy Of The System.
When there is collision between two not perfectly rigid/elastic bodies, Momentum of the system is conserved if there is no external forces acting, same as elastic collisions.
But, if we look at the total energy of the system, before and after the collision, the energy before collision may not be equal to the energy after collision as some energy is lost in deformation of the bodies due to contact or sound or heat, etc. Hence, total mechanical energy is not conserved in this case, but momentum is still conserved.
I think,by 'how much one body touches the other' you mean, how much deformation is happening to each of the body and how do we account for the energy lost in the equations.
To account the energy lost in the deformation, we use something called as coefficient of restitution. it is the ratio of initial relative velocity if the bodies to final relative velocities of the bodies.
$$V_1 - V_2/V_1' - V_2'$$
You can see more about centre of mass and collisions from these lectures, there are really good experiments and explanations in these.