The AGPL requires that you publish a notice that the source is available on demand to the people who use your software over the network. The easiest way to do this is usually to just publish your changes so you can link everyone to it, but that is not a requirement of the license.
You can run derivative AGPLv3 software to service the public without distributing your changes to the source code without violating the license as long as nobody asks for the code.
> The GNU Affero General Public License is designed specifically to ensure that, in such cases, the modified source code becomes available to the community. It requires the operator of a network server to provide the source code of the modified version running there to the users of that server. Therefore, public use of a modified version, on a publicly accessible server, gives the public access to the source code of the modified version.
If you're interpreting that as something different than "publish", I think you're splitting hairs.
That's not in the terms of the license, that's in the preamble as a stated goal. Read sections 4-6. They're not that long and don't really have much legalese.
In practice, the goal is met because someone is likely to request the source for AGPL software. Publishing the code is not a requirement of the license though.
