While copyright is a big area of confusion, a license trumps copyright every time. A license can expand or limit your rights of use, copy, and modification. Our eBooks, for example, are expansive in the sense they are multi-use and simultaneous. All three services, Netflix, Hulu, and Amazon Prime offer a lot of personal content, many of it free, but has some very specific restrictions.
Below are all three services TOS in regards to use of their service.
Unless otherwise specified, the Netflix service, and any content viewed through our service, are for your personal and non-commercial use only and we grant you a limited, non exclusive, non transferable, license to access the Netflix service for that purpose. Except for the foregoing limited license, no right, title or interest shall be transferred to you. http://goo.gl/TFsbX
Amazon grants you a non-exclusive, non-transferable, non-sublicensable, limited right and license, during the applicable Viewing Period, to access, view, use and display the Digital Content in accordance with the Usage Rules, for Non-Commercial, Private Use. As used herein, "Non-Commercial, Private Use" means a presentation of Digital Content for which no fee or consideration of any kind (other than that which you pay to us to view the Digital Content) is charged or received, which takes place in your private home or apartment or, if outside your private home or apartment (e.g., in a hotel room, dorm room, office, or airport waiting lounge) is limited to a private viewing for you and your invitees. Non-Commercial, Private Use specifically excludes any public presentation (e.g., a presentation in a dorm lounge) and any presentation by a place of public accommodation or other commercial establishment (e.g., a bar or restaurant), even if no fee is charged for viewing the Digital Content. http://goo.gl/GlF2f
Your License. Hulu is pleased to grant you a non-exclusive limited license to use the Hulu Services on the Properties, including accessing and viewing the Content on a streaming-only basis through the Video Player, for personal, non-commercial purposes as set forth in these Terms. http://goo.gl/87ihG
All three services in their use of personal or private viewing are clearly stating in their license that you do not have a right to display their content in a public setting. A public setting, as separate from a commercial entity, is displaying content in schools or other community settings. This is why when we purchase DVD's for the library they must have a public license, we can't go to Wal-Mart and purchase DVD's because they only carry a private license.
I don't know if there is a lot of use with these streaming services at school, but you are on very weak legal ground doing so. While copyright itself may be a confusing legal area, licenses that surround it generally are slam dunk cases. A license is a contract you agreed to through use and ignoring it puts you at a degree of legal risk.
A bigger risk in my view are the technological restrictions. Netflix used to have a service with 4 or more DVD's at a time, currently there is only 1. One of the reasons is with the higher DVD's at a time, some libraries used the service for on demand DVD access for teachers. At the district level its easy to see a "solution" to the Netflix issue would be something like lowering the bandwidth to a level where the service would not be usable. This would effect other content besides the streamable services in question.
We have our tragedy of the commons going on. While we have more and more devices connected to wifi, daily there have been connectivity issues in the computer lab. Sometimes programs fail to load, or the desktops freeze because of broadband issues. While the district struggles to solve this problem on the supply side, we all need to be aware that excessive use can effect the connectivity school wide. For me, this is a strongest reason for not using the above mentioned services at school.