Two-Fisted Fighting Training on p. 306 says that it eliminates penalties for using a weapon in the PC’s off-hand.
What are the penalties for off-hand use, normally? I’m not finding them in BWG.
I couldn’t find them either 
The entry doesn’t actually mention eliminating penalties, it just says there aren’t any. 
Minor but I think it matters. It leads me to understand that the skill allows you to do this thing (and notes that there are no penalties for it), and without it you just can’t do it. I guess you could hold two weapons which would reduce the actions needed to switch if you wanted to do that, but that’s about it. I THINK this was the same in Revised too.
So here’s something humorous. This question came up some time ago, and it appears that I chimed in with a similar answer as I did here, and was wrong. I have no memory of this but here it is on record. The short version is, use two weapons if you want but you can’t do any of the stuff that Two Fisted Fighting gives. You basically just choose which weapon you’re using. No Penalty.
I’m going to have to add a sig like Pete’s, just to clear me of any liability from my chronically erroneous participation!
Good catch, Chris. I guess this makes sense, sort of like Shield Training. You can use a shield without it, but are more effective doing so with it.