You're in a tough spot with a mother like this.
At 6 months those kids should be taking a MIINIMUM of two naps a day, each of at least 2 hours in length, or 3 shorter naps. Discuss this chart w/ your MB:
http://www.parents.com/baby/sleep/basics/age-by-age-guide/
Until they get enough sleep you can't really know what you're dealing w/ in terms of temperament. But they are starting to be of an age where they can understand what does and does not garner attention. If they are screaming to get attention (again - not because they're unhappy and sleep deprived, or hungry, or needing a change) then you don't reward that behavior.
But honestly - the sleep has to be fixed.
My twins are almost 3 and they followed the basic rules for average amount of sleep and number of naps as thought they were written specifically for them. They became (still do) totally different children if they didn't get enough sleep.
I also practice the non-attention for screaming thing and it works, but you have bigger fish to fry.
Good luck.