I'm just sure he can tell when there's a threat to him, just like in ROTS when he senses Kenboi is going to kill Vader. I'm also sure he can sense other powerful force sensitives. I could be wrong like always.
When Luke began his training, Emperor Palpatine began to take interest in the fact that the Light Side of the Force was returning. It lessened with the death of Kenobi, but he could sense that someone was being trained.
Give the Twins 20 years of training and Palpatine would have found them far sooner than they were ready for.
Training even Anakins kids would have been dangerous, because they would have had the potential to join the darkisde, perhaps once they learned of their father. In Ep IV, Obiwan took a chance, and in EP V, so did Yoda.
Seems to me that Yoda had begun to rethink taking young Jedi away from their families anmd then training them so early. I think I read that. He felt that the Jedi had become too set in the old ways and that prevented them for finding out about the Sith before it was too late.
i think after Yoda lost to Palpatine, he fell into an extreme set of depression. After being exiled for so long on Dagobah his depession grew. I would be depressed too if I had to live there. lol. I also thinks he looses his Sanity a little. Then again he has been living alone for a long time. lol.
Luke and Leia were split apart to save them. I do not think that Leia was ever supposed to be trained. As for Luke. I think Obi-Wan would have trained Luke sooner if It had not been for Luke's uncle.