They do to an extent.
They are both filmed in Vancouver and share some of the same sets and production teams. This is apparent with places like the coffee shop:
Arrow:
The Flash:
But Oliver Queen's mayoral headquarters and Flash's CCPD building (taken from YouTube) are not the same (notice the windows):
Having a quick look through the IMDb crew lists for The Flash and The Arrow, there's some definite overlaps in the teams as well.
For example, Arrow's cinematography:
Gordon Verheul ... (56 episodes, 2012-2016)
Glen Winter ...
(32 episodes, 2012-2014)
Corey Robson ... (22 episodes,
2014-2016)
C. Kim Miles ... (6 episodes, 2013-2014)
Gregory
Middleton ... (1 episode, 2013)
Shamus Whiting-Hewlett ... (1
episode, 2016)
The Flash's cinematography:
C. Kim Miles ... (23 episodes, 2014-2016)
Jeffrey C. Mygatt ... (11 episodes, 2014-2015)
Stewart Whelan ... (8 episodes,
2015-2016)
Glen Winter ... (1 episode, 2014)
Likewise for film editing in Arrow:
Kristin Windell ... (38 episodes, 2012-2014)
Andi Armaganian ... (35 episodes, 2012-2014)
Paul Karasick ... (15 episodes,
2012-2014)
Jessie Murray ... (4 episodes, 2015)
Thomas
Wallerstein ... (3 episodes, 2015)
Carol Stutz ... (2 episodes,
2015-2016)
And for The Flash:
Chris Conlee ... (10 episodes, 2015-2016)
Nathan Draper ... (7
episodes, 2014-2015)
Harry Jierjian ... (7 episodes, 2014-2015)
Paul Karasick ... (7 episodes, 2014-2015)
Dan Wilken ... (6
episodes, 2015-2016)
So definitely not a complete overlap with identical teams, but certainly shared resources. Looking at the timelines involved, it appears that as The Flash became more established, it got more of its own "dedicated" crew.