SharePoint Online IRM setup

I am new to working with SharePoint so I have been looking online for a walk-through or guides on setting up these settings but appear to have missed something.  

 

I am working in SharePoint Online to setup some protection for company files.  I created a test Library, applied the IRM settings to prevent Printing, require authentication on the file, only allow IRM compliant files, etc.  We did a test and we have a user with access to the file able to save the document locally and it didn't have ant of the IRM applied. Does the local client computer need to have a plugin installed like the Azure Rights Management requires?

 

The end goal would be to lock the files saved in the library down so only authorized users can open or edit the file, lock printing, require authentication so saved files expire and stop being usable, and if possible prevent it from being emailed, copy/pasted to a new file, etc.  

 

Any assistance would be appreciated and thank you in advance!