Note that there is this related question of which this question is by no means a duplicate. It refers to a related but separate scenario.
Long question title, I'll just go with an example. Suppose I developed an image converter program. It takes the filename of an image as input and produces the converted image as output. Now someone modifies my program (by adding functionality to it for example). Instead of distributing the modified version they host a file upload service on a remote server and make this upload service available to the public. What happens on the server is that each uploaded image gets passed as an input to the modified version of my program and the converted image is provided to the user as a download via a corresponding URI. The point here is that while my program has been modified, all (direct) user interaction goes through a separate (upload) service. This service then interacts with my program.
On the other hand the sole purpose why users interact with that separate service is to use the functionality of my modified program which also is producing all output.
Now my question is does the GNU Affero GPL require them to make the source code of the modified program publicly available?
Section 13 of the AGPL states that
[...] if you modify the Program, your modified version must prominently offer all users interacting with it remotely through a computer network (if your version supports such interaction) an opportunity to receive the Corresponding Source of your version [...]
This FAQ states that
In AGPLv3, what counts as “interacting with [the software] remotely through a computer network?”
If the program is expressly designed to accept user requests and send responses over a network, then it meets these criteria. Common examples of programs that would fall into this category include web and mail servers, interactive web-based applications, and servers for games that are played online.
If a program is not expressly designed to interact with a user through a network, but is being run in an environment where it happens to do so, then it does not fall into this category. For example, an application is not required to provide source merely because the user is running it over SSH, or a remote X session.
This sounds like the AGPL would not require to publish the source code in this case. However I also can't believe that this scenario is considered private use of the program. Thus I would like to clarify this point.
In case you believe that the AGPL doesn't help here I would be happy to learn about alternatives which might be effective for this scenario instead.