Here's how I fixed this:
I've had this problem for the last few days, and the only helpful information I could find (other than a bunch of people complaining about this problem) was here
SO, I just did two things. One: a battery pull. Two: Deleted maybe... 10 images that I had taken, and saved on my Galaxy S. I had a TON of free space (something like 11 gig), so I wouldn't think deleting a few pics to make, say... 11.1 gig of free space would all the sudden make the chat app work, so I'm guessing it was the battery pull. It was dumb of me to do both things at the same time, because right now I don't know which worked (although I'm leaning towards the battery pull).
I guess if it happens again I'll try one "fix" at a time.
So yeah... try deleting some stuff that you've saved to your phone (not your SD card) and try pulling the battery, leaving it out for a minute or whatever, then putting it back in and seeing if that works. Try one, see if the chat app works. If not, try the other.
OH, and I found out that chat was working again by signing in to a different gmail account and then sending a chat request to the email address my phone is linked to. Before I did that, the account that's linked to my phone wasn't showing my phone as "online" in the little chat window on the different gmail account, but once I sent the chat invitation to the account my phone's connect to, everything worked as normal.
And yeah, when the chat app wasn't working, I could SEND to people, just couldn't see their text. Weird.