Copyleft only kicks in on redistribution. If you're only testing internally, you don't need to distribute source code. It's only once you distribute that software that you also need to distribute source.
That used to be true, but AGPL changed that. Now it can kick in if you let others interact with the program on your computer.
The IP (intellectual property) of the code has no relationship whatsoever with the IP of the content being served by the served.
> where is the distribution?
You setting up a production server counts as “distributing the service” in AGPL's definition, as AGPL is aimed directly at the SaaS business model.
So if you use an AGPL software on your server, you need to provide the source code (including any changes/customization you've made) to your users. BTW this is a common misconception with GPL, nobody forces you to publish the code in public like on github or anything, it could be behind a login form reserved to your customers only. But you need to give them to your customers under the same AGPL license, so if they want to publish it on github themselves, or fork it and make a business out of it, they can.
Most don't require it, agpl3 does.
(Edit Agpl3)