Yes they both operate to different frequencies.
Two-way radios can be designed as full-duplex systems, transmitting on one frequency and receiving on another, this is also called frequency-division duplex.
How is this not affecting the incoming wave?
Source : Wikipedia
A radio transmitter is an electronic circuit which transforms electric power from a battery or electrical mains into a radio frequency alternating current, which reverses direction millions to billions of times per second. The energy in such a rapidly reversing current can radiate off a conductor (the antenna) as electromagnetic waves (radio waves). The transmitter also impresses information such as an audio or video signal onto the radio frequency current to be carried by the radio waves. When they strike the antenna of a radio receiver, the waves excite similar (but less powerful) radio frequency currents in it. The radio receiver extracts the information from the received waves.
As it says the incoming wave excite but with less power, so it do not effect other incoming waves..
The best example you can find is Transceiver using RF Module
This article can explain different frequency used difference Why do you need two antennas in a radio?