Bill Fitzgerald is someone you should follow on Twitter if you need help interpreting Terms of Service or Privacy Policies. He pays real close attention to technology use in education, especially new, emerging tech.
Periscope is new and teachers love it. Show the Twitters all the great learning that's happening in real time??? Definitely sounds like a good idea.
Please, read this.
Privacy is important stuff. You need to pay attention to how you're sharing information about anyone, including yourself, especially your students.
You can see the whole thread here.
Also, give Bill's blog a read to dive deeper.