Ultimately if you're okay with running AGPL code you're okay with running Mattermost. Which means as long as you don't intend to fork it and not share your changes you should be fine.
That's not how [LA]GPL works. You are free to fork, and not share the changes. The modified source code can be provided only to the people whom you gave the binary (Edit: AGPL has bit more terms, as said in comment below), and you can make it private to the rest of the world (though you can't add additional restrictions).
As Mattermost is dual licensed under AGPL, anyone who uses it under the terms of AGPL can live safe as long as they obey AGPL.
Software pieces using GPL licenses can't add additional restrictions to the software, but of course authors are free to dual license under different terms.
Say for example some one can specify the following for their project: "We provides the project under the terms of MIT for you to modify and run for your own purposes." This might look a harmless statement, but it could mean something very different and may be intentionally written, that essentially makes it non-free. GPL doesn't allow such additional clauses.
Disclosure: IANAL. So I don't know how/if it works in court
From https://www.gnu.org/licenses/gpl-faq.html#AGPLv3ServerAsUser
Of course you couldn't share any derived binaries without a valid copyright license, but for internal use there shouldn't be any problem.
It might work for your company's internal chatroom but not for selling Mattermost as a service.
That's the intention anyway, I'm not a lawyer, I don't know if AGPL was tested in court, and whether it would hold up.
In other words - you can't sell SaaS based on AGPL software + proprietary secret sauce, you have to share back your secret sauce with all users.
It doesn't prevent you from selling it as SaaS, I could sell vanilla Mattermost as a service under AGPL, my selling point might be that I provide high reliability and excellent support.
Do you have an example of that dual license?
I don't see how that would work, the first person you give an MIT license to can re-distribute it under MIT to anyone else, including corporations.
Usually for dual-license you have the opposite, you would have a restrictive license like AGPL for the public, and you would sell non-free licenses with no redistribution rights that allow modifications without sharing back, for corporate clients who want to add their secret sauce to the software without having to give away the secret sauce for free.