Possibly https://obsproject.com/
I can’t vouch for anyone else, of course, but OBS is totally capable of this - I use it this way when doing video chats.
It’s oriented around doing it “live”, though, so probably other software is used by video editors when making videos offline for upload to Youtube or whatever.
You can record offline as well. Just don’t give a streaming key.
Yes. I’ve never steamed live with it, but use it to record videos for YouTube. (Mostly tutorials etc) The ability to use multiple webcams at once etc can be handy.
They do that with editing software. If it’s a still image behind them, then they would need to resize the video to a corner then layer the image behind. It all happens in the editing software. It’s done differently on each program. First layer can be the image where they drag it to stretch along the whole video then the second layer is the video. If it’s another video, then they use a camera that’s pointing at them and at the same time record their desktop (most people use OBS studio) then take both videos and layer them on top of each other in their editing software and resize the video of their face and put it in a corner. Again, it’s all done in the editing software and it’s layered/done differently on each software
For that, I’ve used a green screen.
thats digital compositing. the earliest software on consumer grade computers im aware of was the Newtek video toaster running on Amiga 2000 computers in the 1980s.