The potential scattering of electrons carrying non--zero quanta of the orbital angular momentum (OAM) is studied in a framework of the generalized Born approximation, developed in our recent paper by Karlovets \textit{et al.}, Phys. Rev. A. {\textbf 92}, 052703 (2015). We treat these so--called \textit{twisted} electrons as spatially localized wave--packets. The simple and convenient expressions are derived for a number of scattering events in collision of such a vortex electron with a single potential, located at a given impact parameter with respect to the wave-packet's axis. The more realistic scenarios are also considered with either localized (mesoscopic) targets or infinitely wide (macroscopic) ones that consist of the randomly distributed atoms. Dependence of the electron scattering pattern on a size and on a relative position of the target is studied in detail for all three scenarios of the single--potential--, mesoscopic-- and the macroscopic targets made of hydrogen in the ground $1s$ state. The results demonstrate that the angular distribution of the outgoing electrons can be very sensitive to the OAM and to kinematic parameters of the focused twisted beams, as well as to composition of the target. Scattering of vortex electrons by atoms can, therefore, serve as a valuable tool for diagnostic of such beams. ; Comment: 13 pages, 6 figures