DeveloperB is required to provide all the code that is AGPL. How much of the proprietary code must be re-licensed to AGPL? My understanding is that this is the same as it would be with the GPL, since the license is basically a modified version of the GPL. What changes is that with AGPL the "user" is considered that who runs the program over a network, instead of just the physical user on the machine. In short, any code that can be considered a direct extension of DeveloperA's AGPL code must also be AGPL. If the AGPL code and the proprietary code are combined in a way that they are effectively two parts of the same program, they are considered one program and it must be AGPL. All other code that is "at arms length" with the AGPL code can be considered a separate program and therefore does not need to be re-licensed (an example would be the kernel and a text editor since an AGPL text editor would not require an AGPL kernel despite needing the kernel to function). It does sound vague but I think it's impossible to draw a hard line; it's a matter of defining what parts of a program constitutes a derivative work, or "modified version" as defined in the license itself:
To "modify" a work means to copy from or adapt all or part of the work
in a fashion requiring copyright permission, other than the making of an
exact copy. The resulting work is called a "modified version" of the
earlier work or a work "based on" the earlier work.
For a website's JavaScript, I would say all the proprietary JavaScript built around the AGPL code would probably have to be re-licensed.
Disclaimer: IANAL