I do not understand the meaning of the phase of scattering parameters.
Let's consider for instance a two port network, and let's focus on S21.
I know that its absolute value represents a ratio between power transmitted at port 2 and power sent in port 1. But what is the meaning of its phase?
The phase describes how much the signal is delayed in time from the input to the output.
Therefore the the S parameter can describe how much a signal is attenuated AND phase-shifted in time.
A positive phase means that the output signal is leading the input, while a negative phase results in a lagging (delayed) output signal.
The history of circuit theory tells you reactive components have a phase shift relationship between voltage and current as a function of frequency. As well, scattering parameters also have this complex value.
I think the crucial concept you are missing is the concept of phasor.
A phasor represents a sinusoidal signal of a given (constant) frequency. It is commonly represented by a complex number.
Its magnitude represents the amplitude (voltage or current) of the signal; its argument or phase represents its phase difference with respect to some reference signal (here: the incident signal).
If the incident signal and the scattered signal is represented by a phasor the scattering coefficients also represented by a complex number (though not a phasor) gives the ratio between those signals.
The magnitude of the scattering parameter tells you how much the signal will be amplified/attenuated; its argument (=phase) tells you how much the phase of the scattered signal will be shifted with respect to the phase of the incident signal (see multiplication of complex numbers in polar form: magnitudes are multiplied; arguments are added).
Consider a really simple two port network, a short length of lossless matched cable. S11 = S22 = 0, and magnitudes of S12 and S21 are unity.
The delay through the length of cable will cause a phase shift between input and output. This will be represented in the phase of S21.
