I created a map package in Desktop 10.3.1 and loaded it to AGO. The intention is to allow members of my group to view and manipulate the data. They can see the package, but that's it. They can't do anything with it.
You will want to create a hosted feature service. Take a look at the section 'Publish from ArcMap' in the below link:
Publish features—ArcGIS Online Help | ArcGIS