The GPL and AGPL are two different licenses, and the clauses that require publishing source are very different. The GPL explicitly allows for the request based model, the AGPL (again, exact clause cited below in a sibling comment thread) does not. The AGPL also explicitly requires it be at no charge and be provided over the network.
Edit: Actually I'll just quote the relevant clause of the AGPL here again
> 13. Remote Network Interaction; Use with the GNU General Public License.
> Notwithstanding any other provision of this License, 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 by providing access to the Corresponding Source from a network server at no charge, through some standard or customary means of facilitating copying of software. This Corresponding Source shall include the Corresponding Source for any work covered by version 3 of the GNU General Public License that is incorporated pursuant to the following paragraph.