I have a network game licensed under the AGPL that I modified to suit my needs. Theoretically, I want to charge people to play this game over a local area network that is set up in a private establishment of which I own all the local network infrastructure including the server and the game stations. would I still be required to make the source code available to these people?
It's hard to say for certain.
Based upon the AGPL text, it could be argued that you're propagating the code. As such, you would be obligated to then provide the source.
OTOH, it appears that they explicitly exempt what you're describing in part of their definition for conveyance:
(Update, hat tip to K.Steff!) Section 13 muddies the waters even further.
I would recommend dropping the FSF an email and just ask them. It's the only way you can get a definitive answer. Obviously they have a bias towards saying "yes" in distributing the source, but I think they'll be fair in how they assess your question. It sounds like you're halfway ready to provide the source anyway. Getting a "ruling" from FSF in this case is about as good of a scenario as you can get.
I'm not a lawyer (and I'm not giving you legal advice, please consult a lawyer).
However, it's my current understanding that GPL-style licenses don't care so much about costs and monetary exchange, or public vs private, or ownership of the distribution medium, but more distribution of source and binaries.
The spirit of the GPL is that if you take code from a GPL source,
The AGPL was created to address a perceived shortcoming in the letter of the GPL where hosting some code on an external server that a client connected to was used to shield the source from being released. Ergo, the spirit of this license is that network distribution falls under the GPL constraints.
I personally believe the answer to your question is clear cut based on this information, but again, I'm not a lawyer and there are myriad intricacies that I may not be privy to.
Edited based on comments that responded to my original answer.