Questions about particles and their forces Consider the hypothetical scenario where 2 particles are headed for each other in a collision course but neither interact with a common force. Do these particles simply pass through each other?
 A: Example: photons have not interaction force with each other - and we daily see that one beam of light passes right through another, and does not impact our vision (or the light) in any way.
So yes, they pass right through each other.
A: 
but neither interact with a common force

All particles in the elementary particles table of the standard model have interactions at some level. . All matter as we know it is composed out of these particles. If they did not interact, we would not have studied them.
But particles interact with quantum mechanical interactions, which means that there exists a calculable within the model probability for interacting.
 This can be very very small. Already the two photon interactions are tiny and thus light beams pass through each other  not affecting the optics we have defined. The interaction is very small between two photons because of the box diagram and the diminution of the crossection by orders of magnitude . ((1/137)^2)^2 enters in front of the Feynman integral. This is much worse for neutrinos which interact with the weak interaction  whose coupling is 10^-6 


A Feynman diagram (box diagram) for photon–photon scattering, one photon scatters from the transient vacuum charge fluctuations of the other

These crossections go up with the energies involved in the interaction but to be measurable accelerators are needed.
Thus infinitesimally small probability means in fact that the particles will pass through each other, at low energies.
The rule of thumb is that elementary neutral particles interacting only electromagnetically or weakly or gravitationally (graviton)  in effect pass through each other at low energies .
Electrons on electrons will interact to first order with the electromagnetic force and that has an easily measurable probability of interaction .
Composite neutral particles like the neutron, will interact with the strong force and will be opaque. And the same is true for composite charged as the proton.
