How to distribute private office add-ins?

Eric is partially correct: just 6 months ago, side-loading and private SharePoint catalogs would have been the only way.

However, there is a new "Office 365 Admin Center" that makes it much easier to deploy internal add-ins, and does not require SharePoint. Please see https://support.office.com/en-us/article/Deploy-Office-Add-ins-in-the-Office-365-Admin-Center-737e8c86-be63-44d7-bf02-492fa7cd9c3f