"Colliding" two particles does not mean to bring two classical point-like particles on top of each other.
Even a macroscopic, Newtonian collision of two spheres does not consist of the two spheres somehow occupying the same space - their circumferences "touch", and upon trying to move into each other they repel each other and then separate again. See this question and this question for more discussions what "touching" means, and how things repel each other, but in essence "touching" just means that they get as close as they can under the presence of the repulsive potential between them, and then get pushed apart again.
Now, quantum mechanics makes everything more complicated because particles are not little point-like balls but quantum objects that do not possess a definite position, and the quantum mechanical notion of "shape" is rather different from our classical intuiton, see this question. It is not clear what concepts such as "touching" would mean for objects without a definite position, and so there is no really classical-like description of what happens during such a scattering.
For "slow" particles, a quantum mechanical computation called the Born approximation is typically applied to describe how particles scatter off a target with a potential.
At high energies in colliders, quantum field theoretical processes become more relevant that not only allow for mere changes of direction of the incident particle, but also for the creation of a shower of new particles. The computation of scattering amplitudes that describe the likelihood of different outcomes of such collisions is an essential part of quantum field theory and one of the major experimental tests of our current theories like the Standard Model.