Regardless of whether you are on a channel, in a team lobby or a party lobby, having a profile open doesn't let you select (and gain focus) on the chat box.

The only temp. fix for a shared lobby is to click someone's name from the chat (the one that opens view dota profile, view steam profile in the chat - not the one in your friends list) then reclick the chat box (without closing the context menu; basically clicking the chat box will also close the context menu).

There's a temp. fix for when you're alone too, but I don't think anyone wants to write to themselves :P But if someone does, just click your name (next to the battle points) and then the chat box, before the profile open page appears.

Obviously, closing the profile page also "fixes" the bug.