Independent Analytics creates and stores all of its data on your server. There is no communication with external servers, no personal data is saved, and no cookies are used.
Independent Analytics stores about 200-300MB of data per million sessions. There is also an option to automatically purge old data to limit data storage (from 3 months to 5 years old). For the vast majority of sites, this amount of storage is reasonable, but it could be significant for sites that get millions of visitors.
There is a free version with unlimited tracking and a Pro version with expanded features, such as click tracking, eCommerce integration, campaign tracking, and more.
Yes, you can run Independent Analytics and Google Analytics alongside each other without any issues. This can be a good way to compare stats before fully committing to IA for your website.