Sounds like you need to enable Twitter Tools for your chosen accounts.
Oh, really? I just woke up, so a little caffeine is required, but let's see ...
1) In the Social settings I have the account set to my only Twitter account (set as default).
2) Broadcasting is on by default.
3) No API key is set because @anywhere will be disabled in a couple of months and I don't want to get used to it.
4) Social's comments display is disabled, I want comments to match my theme. When this is enabled I do see a jarringly different comment field.
5) Fetch new comments is set to automatic. This is one of the things that does not work.
6) Debug mode is on, but debug_log.txt is empty.
7) My WordPress User Profile is also configured to my only Twitter account (set as default).
8) In the Twitter Tools settings the show admin screens for tweets is set to yes.
9) Create URLs for tweets is set to yes. The actual URL is different from the example, at least with the tweets converted from prior to TT3.0.
10) TT3 credit link is active.
11) Debug logging is enabled.
12) Account configured in Social is no longer greyed out and the enable Twitter Tools option is checked.
13) Create blogposts from tweets is checked (I hope this only creates the special tweet pages and not a real post on the main page). Retweets are excluded.
14) Both the Social and Twitter Tools plugin directories are set with ownership of root.apache and permissions of 775.
15) SELinux has been switched from enforcing mode to permissive while I troubleshoot this, but this plugin shouldn't be doing anything that Apache, PHP and MySQL can't already do.
That's everything that was set before I went to bed last night, so what have I overlooked?