There’s an old saying that we have one mouth and two ears so you can listen twice as much as you talk. However, talking and listening at the same time is fairly difficult and doing it with radio signals is especially hard. A company called Kumu Networks has an analog module that can use self-interference cancellation which allows transmitting and receiving on the same frequency with around 50 dB of the transmitted signal in the transceiver. You can see a video about Kumu’s claims its technology below.
You may think that cell phones and ham radio repeaters transmit and receive at the same time, which of course they do, but usually on different frequencies to avoid direct interference. A diplexer is a device that sorts out the two frequencies while a duplexer sorts them out by the direction of the signal, but they are tricky to use. A duplexer can operate on a single frequency in applications such as radar, and even then it is still very difficult to prevent leakage from the transmitter from overloading and desensitizing the receiver.