As to why it does not work, see which way does the current have to go to charge each input cap,
you will find it needs to go thru the opposite side regulator and load...
Read up on the concept of "virtual ground". You need a way for current to bypass the side that requires less current.
Fix could be as simple as putting two ~8V zeners across the input caps to bypass the other side unloaded regulator. Make sure that together the zener voltages are slightly higher than your supply voltage.