This is just a minor suggestion for a QoL feature: I would love to be able to make the chat input (currently limited to one line) larger, perhaps by dragging the line separator between the chat input and output upward, so that more input text is readable at a glance before submitting it.
Alternatively, the input area’s size could also grow vertically depending on how much text has been entered, perhaps up to a limit of around 7 lines (or a dynamic cap relative to the browser window’s height).
… and just after I posted this, I noticed there is an arrow on the right side to expand the chat area! Oh well! 
Making the size grow dynamically may still be a nice feature, though!
There is? Could you attach a screenshot? The only chat arrow I’m aware of is for in game canned messages…
That would just encourage people to talk too much 
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But can you really talk too much? one thing i would add on to this suggestion is to make the chat tabs appear from left to right as opposed right to left.
The reason i would suggest this is that often, the chat window, blocks other chat spaces, and because the messaging service on OGS isnt the best ergonomically, like you would expect to find on a messaging service like on Facebook, this small change would make life easier… also the ability to move the tabs would be great too I think.
Also the message length cap is a little small for me, i can get pretty wordy.
But that way, any new PM would block a part of the board when in game. I am against that. I think that being a Go server the game itself should always be prioritized. If you want to chat, you can take the extra step or click, but if chat starts interfering with the game (especially since it is often a time sentsitive thing), that’s not good.
You can drag and drop them already
Only when they’re open.
When minimised they snap to bottom right corner.
Its hard not to agree with what you said to be fair. Alternatively though, having a chat tab appear on the screen can be quite distracting… though the only real idea i would have for that would be to make the tabs not appear at all, but a notification to appear instead so that you can open the chat tab at your leisure.
But at this point, is really making or even suggesting these things at all important at all when as you rightly suggested the game should be the thing that comes first.