quantum mechanics - Is there scattering event if two wave functions overlap in momentum space?

Let's consider two particles, moving, for simplicity, in a 2D plane, represented by their wavefunctions in the form of (when being modulus squared) very very very narrow Gaussians (let's say close to the infinitely narrow ones). So, it means that the positions of the particles are quite well defined. And according to the Heisenberg's uncertainty principle, the momenta of two particles are not well defined. The last means that wavefunctions being represented in momentum space have very very very large width of the corresponding Gaussians. And in momentum space they are definitely overlapped. Two wavefunctions, being overlapped, interact with each other. So scattering event in momentum space should take place, even if in the real space they are not overlapped. If there is scattering event in the momentum space, then it means that momenta of two particles were affected, which implies in turn that something should change in the motion of the particles in the real space. Am I right? It means in turn that particles can scatter from each other even they "don't touch" each other... Where am I, if I am, wrong in my thinking flow?