Hello 10b
I would pile the weeds separately for composting. Think of that as a long-term composting project. Pile, let them rot down, turn, repeat in a month or two, or even six. If there are viable seeds, roots, shoots, etc. in that pile, you want to have a good long period of time for that material to completely rot before using it. I have one large bin with an open base that I keep adding weeds to each year. when it is full, I will let it rot down a while without adding more, then fully turn it over and start a new bin next to it.
As for the fresh manure, these days I usually only get get compost with manure as part of the blend. It is fully composted manure. I don't have fresh manure available to me right now. However, it seems like your plot neighbors have the right idea with putting fresh manure on top of the beds, letting it over-winter, then turning it into the soil. You can compost additional manure that can be used after the composting process is finished.