You can check and make sure that they have the necessary permissions on the object that houses the data. It is highly unlikely that the user can see the data but cannot manipulate it unless the security settings have been set to Read Only for that profile. Edit the profile and check under Standard/Custom Object Permissions. If you are using a default Community User Profile then clone it as SF does not allow you to change some settings in the default profiles as a protection for keeping a standard profile. You can also check the permission sets and if there are any triggers set in the background preventing the user from performing the export. Hope this helps?
For Communities, main thing you will need to do is to find the ExportData Visualforce Page included in the Skuid package, and then make a local clone of it, as described in the "Prerequisites" section of this tutorial.
Once you have created this Visualforce Page, you will need to assign this Visualforce Page to your Partner Community Profile(s).