I noticed there really isn't any data on the changes to the circle abilities, so I decided to do some testing of my own, perhaps to jump start more-methodical players. I am aware that my tests are not perfect, but the mathematically inclined may be able to get some rough numbers out of them. My tests included 5 uses of Arcane Circle each, for a total of 7.5 minutes for each test, and I did not have signet for either of these tests.
Test #1
Job: DRK75/RDM37
Weapon: Warp Cudgel, capped fSTR (24 DMG) (note 25% damage bonus due to blunt)
Stats: 250 attack, 283 Defense, 66 Vitality
Enemy: Level 56 Dromas
Range of damage dealt (Without Arcane Circle): 31-51
Range of damage dealt (With Arcane Circle): 36-62
Range of damage taken (Without Arcane Circle): 26-59
Range of damage taken (With Arcane Circle): 21-45
Test #2
Job: DRK75/RDM37
Weapon: Seiryu's Sword, capped fSTR (58 DMG)
Stats: Capped cRatio
Enemy: Level 41-43 Clockwork Pods
Range of damage dealt (Without Arcane Circle): 91-139
Range of damage dealt (With Arcane Circle): 108-156
This second test is interesting, as you can see that the Circle effect raises your effective DMG rating as well as possibly raising attack directly.
I hope these tests give a decent enough starting point for more exacting tests.