For me, it depends on the "weed," the time of year, and the state of the bed. Typically, grasses, morning glory, and anything else that I deem to be "aggressive" is pulled & tossed on the surface or paths. If it's a wet/humid time of year I will cut/tear the
roots from the stem since the weed will just keep growing if I leave it intact. When it's hot/dry I don't worry about it because the sun will dry it out before it can start growing where I toss it. If they've already gone to seed, I usually pull & toss outside of the garden. Iwill also pull weeds if the seedlings or transplants are still too small to compete.
For things that are unlikely to grow through the mulch, I just cover with more wood chips. I also cover with mulch if the crop plants are already big enough that I won't mistakingly smother them with mulch. If the weed has already gotten fairly large, I may cut/break the top off and smother the roots with more mulch.
Lately, I've been trying to dry the weeds like
hay or straw by laying them over the small piece of fencing I put up for beans to climb on this summer. Since I haven't been able to get any more chips delivered lately, I'm trying to utilize all my sources of biomass for mulch, and letting them dry out helps minimize the chance of them re-sprouting after I apply them as mulch.