Agree with redhen and purpleroses about the importance of having very very clear house rules.
For me, there is a key distinction between the things which violate house rules or which otherwise impact me and DD, and the things which I might dislike or disapprove of, but which have less direct impact on my homelife or DD's life.
So we have 'house rules', and these are agreed by me and DH and made clear to the kids. There are consequences set for breaking these, generally loss of pocket money, grounding, or loss of electronics.
I will tell the kids when they need to do something, but when it comes to the disciplining, DH 'delivers' the news, usually - although in a few cases of extremely bad behaviour, I have been the one to dish out some consequences!
I should add that my DSCs were already 10 and 13 when we met, so I am more hands-off than I would have been with younger kids.
For the things that do not have a direct impact on me or violate house rules, but which I feel are not good for the kids - who I care very much for and so want to see then disciplined as well as loved - I try to be a support and sounding board for DH. I do tell him what I think and how I see things playing out...we often end up talking through various options until DH finds one he feels right about. Without me, he'd always choose the path of least resistance, so I know I do have influence on these things.
My DH finds it hard to discipline, speak firmly to, or even allow 'natural consequences' to affect the kids. He is a very 'Disney Dad' and was even when the kids were small, just gotten worse after their mum passed away. He also used to favour one of his kids a bit when it came to getting away with things, which he has turned around very sucessfully. He's been trying to change his approach and things are lots better with the kids than they once were. He knows that he wants things to be different, and just needs support getting there. He is not always consistent, and the hardest part is that he tends to be a bit 'checked out' sometimes so he doesn't always hear or notice things that he really needs to - that is probably what we argue about most.