Screen or Tmux? Both

Why use one locally and a different one remotely? Different default bindings? You could change them.

Or is it that tmux might not be available remotely? I'm sure you could build a static binary and dump it in your homedir.

Which tmux also has (tmux detach / tmux attach).

...and sharing...

Which tmux can also do, arguably better, because it doesn't require a setuid binary.

If you need to create multiple console windows inside that server connection, then Screen allows you do to that as well.

Which tmux also allows.

Having said all that, like the author I also use tmux locally and screen remotely, but this is more a function of the relative newness of tmux, and ubiquity of screen than of any extra capabilities of screen.

I do this, simply ctrl-b is just a pain combination to input constantly compared to cntrl-a.