The transmission through a two-dimensional (2D) channel, which connects semi-infinite 2D electron gas reservoirs and can be used as a model of both the few-atom-size (microscopic) and mesoscopic contact is investigated under the assumption of the zero-field ballistic motion of electrons in the system. An analytical theory is advanced to a nearly exact analytical expression for the transmission probability verified by numerical calculations for a vast variety of channel parameters. On these bases a detailed analytical and numerical analysis of size effects in the quantum point contact (QPC) transmission is performed with emphases put on basic distinctions between atomic-size and mesoscopic-size QPC. The QPC properties are visualized in calculations of transmission against Fermi energy dependences for some representative channel parameters. The results obtained are directly applicable to the description of coherent excitation energy transfer in relevant molecular 2D geometries. The suggested formalism can be also used for an analytical analysis of some other problems of current dispute, in particular, of non-linear field effects and 3D QPC properties.