In high energy experiments, people smash particles into particles.
But how to make sure that they really run into each other, instead of just passing by?
You're using beams of particles, and you focus each of them as much as you (practically1) can so that the particles in each beam are reasonably close together.
The result is a wide variety of interaction distances from far apart through near misses to closer interactions still. You mentioned electrons which are treated as point particles so that every interaction is a "miss" at some level. For hadrons (protons, neutrons, mesons, etc), alphas and heavy nuclei the constituents of the beam have physical extent and can reasonably be said to collide at times.
To recap: the particles in the beams necessarily interact with each other at a variety of distance.
Now you've asked the critical question. The answer is that we design and operate the detector package to measure enough data about scattered particle to reconstruct that information.
A simple example that was used for decades is to simply only place detectors at positions that represent large scattering angles (which implies high momentum transfer, and therefore very close approaches or actual collisions).
As data acquisition systems have become faster it has become more a more feasible to simple collect everything and sort the details out later.
I've written the above assuming a beam-beam machine, but similar remarks apply to fixed target work.
1 There are good reasons not to just get the tightest focus that can be achieved. In particular the tighter the focus you attempt to create the move transverse momentum you give the beam particles and the harder it is to contain the beam and bring it around for another go.