What do we mean when we say that it requires the particle nature of radiation i.e., photons, to explain photoelectric or Compton effect?
Both the photon and the electron are point particles in the elementary particle table.
We cannot say that the momentum and energy are distinctive properties of a particle. Waves, too, have these properties.
Classically a particle is described not only by energy and momentum but also by a specific center of mass point at (x,y,z) which allows to determine a unique trajectory given the initial conditions. If it is a point particle more so. Impact points are also clearly known given the trajectory. A classical particle hits a wall at a specific (y,z) point.
Neither of these two processes ( photoelectric and Compton) is good for displaying the particle nature of the elementary particles, because their detection depends on the probability distribution measurable by the accumulation of many events.
The photoelectric effect demonstrates that photons exist as individual energy packets and not a continuum classical wave carrying energy. The same with Compton scattering, where the photon gives off part of its energy to the electron. But both cannot be used as a clear demonstration of a classical particle behavior, because both depend on probability distributions for the trajectories, there is no uniqueness of trajectory for the same boundary conditions.
IMO the best teaching demonstration of the dual nature of elementary particles is given by the double slit experiment.
The particle nature is the footprint of each individual electron , which is a unique x,y point on the screen, as expected by a particle trajectory. Its position seems random , until the wave nature is displayed in the accumulation. It is a probability distribution which is describable as the complex conjugate square of the wavefunction describing "electron scattering on two slits"
A better demonstration of a particle nature is this bubble chamber picture of a pi mu e decay
The pion from the main interaction decays to
Then the muon decays :
The track until the decay point acts like a classical trajectory , at that momentum and the same magnetic field it will have the same circular track. At the interaction point the probabilistic nature enters, which is tied up with the wave nature of the elementary particle: there is a probability distribution for the distributions of how the mu+ and nu_mu share the momentum energy, and the same for the sharing of the e+ nu_e and anti-nu_mu in the second decay, and it is probability distributions that display a wave nature.