Do “noise-canceling” headsets work well for players sitting next to each other?

So, my friend and I have two ps4s set up next to each other and like to play coop in this way. However, if we don’t use microphones to communicate we can’t hear each other well due to the game sound in our headsets. And, if we do use mics we can hear both the sound from the headsets AND our voices in the room, with a slight delay over the headset. So far it’s a headache and I end up playing with one side of the headset slightly moved away from the ear.

Question: Are “noise-cancelling” headsets able to solve this issue, so that we would

a) be able to sit next to each other,

b) be able to communicate using mics,

and c) not hear our normal “in-the-room” voices at the same time? Any recommendations? I’ve googled, but cannot find the answer to the specific question above.