The Hand In share is intended to allow students to submit work to the teacher without having to print it out or email it in. The Hand In share only allows students to copy their files into it. They cannot save work directly there (by using "File -> Save" in Microsoft Word, for example), nor can they read, alter, or delete any documents in there. To hand in an assignment, they should open two windows on the computer: one for the Hand In folder where they want to copy their work, and one for their own home directory (H: Drive or My Documents). Then they can just drag their work from their home directory to the Hand In folder.
Teachers in the school should all have full permissions to create files, read files, move files, and delete files in both the Hand In and Hand Out areas.
The Hand Out share is intended to be used by teachers to give students a copy of a document that they should work on. Students can open files here or copy files from here, but they cannot save anything here, or modify any files here.
Remember that this space is shared by all teachers in your school, so you should make an effort to keep it tidy. Create a separate folder for your class to submit their work to, and clean up any files that get put in the wrong spot (at the top level of the Hand In folder, for example). It's also a good idea to copy files from Hand In to your personal home directory, as it is possible that another teacher could accidentally delete your folder on you.
I hope this helps to explain how these two shares work!