How is the beam-beam tune shift parameter measured?

The beam-beam parameter (or beam-beam tune shift) cannot be measured directly.

It can be inferred from the luminosity by inverting the relation between luminosity and beam-beam tune shift (usually for the vertical beam-beam tune shift).

It can also be calculated from the basic definition by plugging in the beam and optics parameters.

In certain cases, for simple collision patterns (nt good with multiple bunches) it is possible to observe the beam-beam tune shift on the transverse tune (frequency) spectrum of the beam. One can consider a pier of colliding bunches like a system of coupled oscillators, the beam-beam force playing the role of the coupling term. One then gets 2 main oscillation modes (eigenmodes) called the sigma (bunch oscillate together) and pi mode (bunches oscillate in opposition). The frequency difference of the 2 modes is proportional  the beam-beam tune shift (factor not too far from 1...).