So, if I understand the license correctly, anything you produce with the product becomes AGPLed and distributable to the users? That’s crazy. That’s like saying every Word doc I produce is now AGPLed. Or do I misunderstand something?
The AGPL is identical to the GPL, but with one additional clause: if you use the modified software to provide a networked service (like a website), then you must distribute your modifications. That's all.
You can use AGPL'd software to produce content without fear that that content will be encumbered, since you're not providing a networked service through that content. That's why your documents are safe, even in the (unlikely) event that your word processor is AGPL.
That depends on what you mean by "detect" -- there's an large body of Computer Science research dedicated to program attestation, i.e. enabling a user to verify that the program (or results of a program) derives directly from some source. But it's far from a solved problem in the general case.
Less formally, that's what the legal system is for: if the copyright holder suspects that the published source code is different from the source code that's being run by the service, then they're perfectly within their rights to take the service provider to court and attest, under the penalty of law, to their compliance.
The law
>Would that be easy to detect?
No
Modifications to views, models, stylesheets, &c. would all fall under the AGPL's definition of source code and would be considered sufficient for triggering the distribution clause.