Perfect fluid tori with uniform distribution of the specific angular momentum, $ell(r,theta)=mathrm{const}$, orbiting the Kerr--de~Sitter naked singularities are discussed. Closed equipotential surfaces corresponding to stationary thick discs are allowed only in the spacetimes admitting stable circular geodesics. The last closed surface crosses itself in the cusp(s) enabling outflows of matter from the torus due to the violation of hydrostatic equilibrium. The inner cusp enables the accretion onto the ring singularity. Influence of the repulsive cosmological constant, $Lambda >0$, resides in the existence of the outer cusp enabling the emph{excretion} (outflow of matter from the torus into the outer space) and gives rise to completely new type ofa disc called the emph{excretion disc}. The plus-family accretion and excretion discs can be both the corotating or counterrotating discs, the minus-family ones are always the counterrotating discs, as related to locally non-rotating fra