I don't have nearly as much stepmothering experience as MaryP
, but my husband and I operate in a similar fashion with his kids, who are with us part-time. (My son's older and out on his own, and that's yet another dynamic...)
When we first moved in together, I would discuss issues involving the kids with their father, but generally kept my mouth shut about addressing anything with them directly. When I did, it was usually because their dad wasn't around to see what was going on, and I'd speak in his name. After their dad and I got married I got more comfortable about dealing with things myself - probably because I had "official" step-parent status at that point, and therefore more authority. (That's probably just my take, though - I don't think it really made, or makes, much difference to the kids, since I have yet to experience one of my biggest worries - hearing "I don't have to listen to you. You're not my mom."
On big issues, I do think it's up to the bio-parent to deal with the child. I'll discuss things with my husband and support him, but I think it's more important for him to get on the same page with their mother than with me in those cases. On smaller matters, I've become much more outspoken, especially when I notice misbehavior before their dad does. The funny part is that despite the big worry I mentioned, I actually get less
backtalk and more
prompt responses from the kids than their dad does.