  In this paper, quantitative central limit theorems for $U$-statistics on the
$q$-dimensional torus defined in the framework of the two-sample problem for
Poisson processes are derived. In particular, the $U$-statistics are built over
tight frames defined by wavelets, named toroidal needlets, enjoying excellent
localization properties in both harmonic and frequency domains. The
Berry-Ess\'een type bounds associated with the normal approximations for these
statistics are obtained by means of the so-called Stein-Malliavin techniques on
the Poisson space, as introduced by Peccati, Sol\'e, Taqqu, Utzet (2011) and
further developed by Peccati, Zheng (2010) and Bourguin, Peccati (2014).
Particular cases of the proposed framework allow to consider the two-sample
problem on the circle as well as the local two-sample problem on $\mathbb{R}^q$
through a local homeomorphism argument.
