When you aren't logged in, on the bottom of the screen there is a link that says "logged in" - this really confused me at first as I assumed, since I log into SE with Google, it automatically logged me in via my Google account (it is Android after all). Problem was... I couldn't figure out how to chat via my phone! I felt silly and even downloaded a different browser thinking maybe it was just a rendering problem since the menu button looks like its sitting on top of a text box.
I soon discovered this was a typo and that the "logged in" text was a link to the log in page so it should say "Log in" or something to that extent. (Once I logged in I could chat just fine)