Voice communication between ground and spacecraft during missions such as Apollo seemed to be always punctuated by beeps. What was the origin of these beeps? Were they some sort of unavoidable artifact of the communications system in use? Did they have a purpose as a sort of automatic "Over!" as is customary in voice communication on simplex radio? Was it to help the communications equipment switch over between transmit and receive modes? Was it some sort of feedback occuring when both parties had their microphones keyed simultaneously?
Those were called Quindar tones and they were used for in-band signaling in the ground communications network to key the transmitters at the ground stations. An "intro" tone is generated when CapCom in Houston presses his/her push-to-talk (PTT) button. This triggers the high power amplifiers at the MSFN ground station (often on the other side of the world) to start transmitting. When CapCom is done talking, the PTT is released and the equipment at Houston generates a slightly different "outro" tone. The equipment at the ground station recognizes that and stops transmitting.
As Michael Kjörling mentioned, comms were full-duplex.