Everything I've ever read about it said it was no problem and not an indicator of readiness to be trained again. On most routines where I trained every bp twice or thrice a week this was a necessity, at least in the beginning stages. My strength never suffered but my overall feeling during and after a workout and my pumps took a hit for the worse. Then you have to also factor in that the less often you train a bp, the more sore it gets and the longer it stays sore as well.
In short: the consensus is always that it's just fine, personally I'm still in the middle about this, I honestly don't know how much and if any truth there is to both sides
IMO training muscles hard that are still sore is a bad idea. I tried it once in college and went on for a month or more like that. I ended up losing strength and didn't put on any muscle. I don't think I lost any though.
I think you can train the muscle but I would use lighter weight/high reps. Do sets of 20 and just try to get blood in there. Don't go to failure. Goal should be to pump lots of fresh blood in there and flush out metabolic waste. It should even help to speed recovery if you do it right.
There was something guys used to call "feeder sets". Youd do those the day after you trained hard. Just a set or two with light weight not done to failure. Pump the muscle up with lots of blood.