How do I switch between desktops and full screen apps when accessing a remote machine via Screen Sharing?
Will have to go into system preferences, keyboard, shortcuts, and then assign a specific key to activate Mission Control. You can do this either in your host machine or in the remote one. What matters is that the shortcut for the remote machine is not used in the local machine. 

You can do the same directly for the ^right and ^left shortcuts. Change them in one of the two machines and then they will work in the remote one. 

For example I indicated the F10 to be mission-control on my remote computer, so whenever I press FN+10 I'll get mission control on the remote Computer