There used to be a christian newspaper called Challenge Weekly in nz. I think it was non denominational, but it ceased publication about a year ago.
Then another started up called christian life, I think. They were freely given out at christian bookstores. However the mega churches tend to advertise in that one and its not really a church paper...its kinda of a free for all.
The only other one I know is NZ christian writers, but thats more of a journal than a magazine.
When the baptist mag used to be a paper it had births, deaths, marriages and baptisms.,it had letters and obituaries, and church reports around the country, testimonies, missions, etc. it wasnt just articles about church leaders or navel gazing ones, like 'what is baptist identity?!'it was about what was going on in all the different baptist churches and encouraging articles about faith in action.