GNU is very clear on that: Under the GPL, you are not legally required to offer the modified source code. The word "legitimate" in the second paragraph of the FAQ entry is a bit misleading, but they rather mean something along the lines of "reasonable, understandable".
There is a special license, the GNU Affero GPL, that handles this question differently:
The GNU Affero General Public License is designed specifically to ensure that, in such cases, the modified source code becomes available to the community.
There you also find a clear note about the normal GPL:
The GNU General Public License permits making a modified version and letting the public access it on a server without ever releasing its source code to the public.