Onlyoffice desktop editors is now free software under the AGPL license.

"I'm confused. AGPL is for server-side software, right?"

Now that things like https://www.rollapp.com/apps exist where you can hide GPL "desktop" apps behind a website it no longer makes sense to distinguish this. Using the AGPL is always going to be better, even for programs that you might not think would ever get used over a network connection, simply for the RollApp idea.

"Just had a more careful look at the website and realised they offer web-based office functionality as well"

They've got free and proprietary versions. The free version under the AGPL is no doubt to provide advertising to get suckers -- er people -- to buy the proprietary version (because of their "Pursuant to Section 7(b) of the License you must retain the original Product logo when distributing the program" - Note that this has no conditions on it, so even if it's been modified.)

Also note that the company only makes money when buying the proprietary version so they're clearly betting on people doing that. Given that they have a financial motive I wouldn't be at all surprised if they started doing AGPL enforcement in the style similar to MySQL AB where they'd shake down people over things that are arguable violations and say "You should buy the proprietary version because Geee... that sure would be sad to see you get sued over that..." (MySQL AB for example made crazy claims like embedding SQL code was a violation and etc. so I wouldn't be surprised to see a company with a similar financial motive to make other similar claims and have people settle by buying the proprietary license because of fear of being sued.)

It also seems to act as trialware for their proprietary version. Surely if you sign up for "their" hosted "cloud" version you get the proprietary version with all the features, but if you want to do it yourself you have to go with the watered-down version with less functionality. That also helps to show that they have things set up in such a way as to steer people toward the proprietary version.

Also note that it's licensed under AGPLv3 ONLY. Not only does that create a future compatibility problem, it is presumably done that way because they don't know what version 4 will say and if version 4 might be changed in such a way so as to somehow undo or break the "scheme" they've got going on here with their watered down trialware version that they're putting out in the hopes that they'll be able to convert them over to a sale of their proprietary version (and if not, that least they get some free advertising because of the requirement under 7(b) to keep their logo there regardless and then someone ELSE that sees it will go and get the proprietary version.

As a result, while it's free software, using their Community Edition seems like it's REALLY only there to prop up their proprietary software business which I find distasteful.