Chat between two people sharing byobu screen

Absolutely! This we do this all the time. There's a couple of little, lightweight options:

1. On a command line, just start with the shell comment character, "#", like: # hey, dustin here, what are you doing with that file?
2. Open up a text editor like vi(1) or emacs(1) and start typing
3. Use the wall(1) command to talk to all windows everywhere

Full disclosure: I am the author and maintainer of Byobu.