I don't know enough about the admin side, but perhaps it's something that needs to be enabled separately. From a user perspective it 'just works' in our office:

If several people open a document shared on OneDrive, that document is opened in a collaboration mode where you can see others working on it, similar to Google Drive.

A menu appears at the bottom of the screen[1] showing who's editing that file. If you have Lync/Skype for Business, clicking on those names opens an IM to that person.

Re privacy, a quick search shows their cloud services are compliant with ISO 27018 as of February this year[2]

(and no, I don't work for Microsoft!)

[1] http://core0.staticworld.net/images/article/2013/04/who-is-e...

[2] http://www.zdnet.com/article/microsoft-adopts-international-...

Thank you!