I don't know the rate, but you can still get new skills from the LDs. But I think you have more of a chance if you have been using a character longer, if that even makes sense. They made it so you can't get duplicate skills, but it also seems like the droprate is less.
The droprates fro LD's are very inconsistent. I've gone months not getting any, but then got 3 skill drops over the past two days from farming Crawler during this point bonus event.
I read it was about 0.5% per drop against LDS (i.e. per 1000 points scored) and unlike SR/UR card drops it doesn't depend on the LD's level so 10 and 40 have the same chance of dropping a skill.
It may be a different in PvP duels (I suspect it's better) and it may depend on how many skills there are left to get.