  We study nonlinear transport and non-equilibrium current noise in
quasi-classical point contacts (PCs) defined in a low-density high-quality
two-dimensional electron system in GaAs. At not too high bias voltages $V$
across the PC the noise temperature is determined by a Joule heat power and
almost independent on the PC resistance that can be associated with a
self-heating of the electronic system. This commonly accepted scenario breaks
down at increasing $V$, where we observe extra noise accompanied by a strong
decrease of the PC's differential resistance. The spectral density of the extra
noise is roughly proportional to the nonlinear current contribution in the PC
$\delta S\approx2F^*|e\delta I|\sim V^2$ with the effective Fano factor
$F^*<1$, indicating that a random scattering process is involved. A small
perpendicular magnetic field is found to suppress both $\delta I$ and $\delta
S$. Our observations are consistent with a concept of a drag-like mechanism of
the nonlinear transport mediated by electron-electron scattering in the leads
of quasi-classical PCs.
