Well, there was overlap in staffs. Primarily at the higher level, though. Ron Moore and Brannon Braga wrote for TNG, DS9 and VOY. And I'm guessing there were at least a few others who wrote for the latter two series.
Well, Ron only came onboard Voyager in season six, after Deep Space Nine finished its run. After writing only two episodes, he promptly left.
Braga never wrote for Deep Space Nine, according to Memory Alpha. I believe he jumped straight from The Next Generation to Voyager (perhaps with Generations and First Contact in between).
Yeah, Berman was the only unifying force... and Moore wrote this 5 page thing on why his Voyager stay was short lived and what he thought was wrong with the show.
It seems like the "franchising" was the split in the family.