I like to pull the weeds out before laying down the newspaper or plastic sheeting. Invariably I have seen weeds eventually growing through the newspaper (which breaks down into to soil and is a good thing) and that plastic sheeting.
Be careful with plastic sheeting and find out what polymer it is. If it si PCBA and PVC do not use it. PCBA is very toxic and leaks into the environment, PVC itself is quite stable and inert however, it is also very brittle and plasticizers are added to it to make it more flexible and they leak into the environment (and our drinking water
Anyway, my initial point was that I like to go over the weeds first and pull as many of them out as possigble by hand or by using a three pronged or stirrup or dutch hoe. (and there are others). I put the weeds into the compost pile, trench or other and then lay down my black and white newspaper.
Then, if I'm not planning on planting anything else there, I lay bark mulch over top of the paper. Works well.