I highly recommend Open Broadcaster Software (OBS). It’s free, open source, and runs off your GPU which means the performance hit is very low (less than 5fps). It uses a source system that allows you to record individual windows, whole display(s), and allows you to arrange multiple re-sizable sources on top of each other, or any other layout you want. It’s a really powerful tool, but the interface is relatively simple and easy to use.
I’ve been using it for a long time now to create Youtube videos, but it also has built in streaming mode that is easy to hook up to streaming services online.
Here’s their site: https://obsproject.com/
EDIT: Just noticed this thread got necro’d… hopefully this is useful to someone else 