2. The vatman and the coucher
Sheet forming’s key man was the vatman. He used a mould formed of a bronze wire mesh and a wooden frame to cover it called a deckle.

A small modern mould with deckle raised above it
He would dip the mould plus deckle into the vat of pulp and water, allowing water to drain out leaving a consistent layer of fiber on the wire mesh. “The vatman spreads the stuff on the mould by gently shaking from right to left and from left to right, as if he wished to riddle it, until it is spread equally over the whole surface of the mould; this is known as “promener”: or to “shake”. In the same way, by another movement which is made by pushing the mould forward and pulling it back in a to and fro motion, as if riddling, the stuff binds and knits together and becomes perfect; this is known as “serrer” or to “shut” the sheet. These two movements are accompanied by a slight shake which serves to “put together” the sheet, that is to say to fix and bind it; but they are carried out very quickly with seven or eight movements of the hands and in the space of four or five seconds. Immediately this stuff, so fluid, which seems no more than slightly cloudy water, knits together. . . . In this way the sheet precipitates onto the brass screen while the water drains away through the interstices and a real sheet of paper remains on the mould.” (Joseph-Jérome Lefrançois de Lalande, “Art de faire le papier,” in Description des arts et métiers, vol. 4, Paris: Académie royale des sciences, 1761. Translated by Richard MacIntyre Atkinson as The Art of Papermaking, Kilmurry, Ireland: Ashling Press, 1976.) This action is one of those “flow” activities which it took years to learn: the vatman was responsible for the consistency of thickness and weight of each sheet, for its formation — ideally consistently even across the entire mould, and for the speed at which the whole operation could run. The vatman would work with two moulds and one deckle, handing off the first mould after dipping one sheet, and then putting the deckle onto the other mould to form a second sheet while the coucher was unmoulding that first one. (This all becomes much clearer in the two videos attached to the post Paper making by hand 4.)

from Schopper: Panoplia omnium illiberalium mechanicarium . . . 1568
This illustration from 1568 is the earliest known printed illustration of paper making.It was drawn by Jost Amman. The vatman can be seen about to immerse the mould in the vat of pulp, with stamping mill and the drying press in the background, and the water wheel driving the whole operation outside the windows. (The lad carrying that stack of paper is either Superman, or fanciful.)
Working right next to the vatman, but not illustrated would be the coucher. His job was to form a stack of (very damp) sheets. He would take the mould, without the deckle which the vatman retained, and carefully invert it to allow the sheet to drop off onto his growing stack (post) separating each sheet with a felt to absorb more moisture. From the moment the vatman takes the mould out of the vat, paper making amounts to removing water without disturbing the fibers. After the coucher has built up a post it would be passed on to the layer who would begin building another stack, this time of paper only. He would take each sheet off the felt in the post and transfer it to a separate pile, handing the felts back to the coucher.

From left to right: layers, coucher, vatman. (From French Royal Academy of Sciences encyclopedia of occupation, 1761)
The layer needed to be able to build a neatly aligned stack with a quick motion, not tearing the corners or leaving fingerprints. After they had accumulated enough sheets the stack would be put into a screw press in order to extract water. The press can be seen in action in the lower left hand corner of the engraving above.
If the paper was to have a watermark, this would be incorporated into the mould. The illustration below shows a mould from Hayle Mill with a quite elaborate wire watermark woven into it. (The video of Hayle Mill shows them making watermarks.) The watermark simply makes the paper a little bit thinner in the areas where the extra wire is placed, thus making the paper a bit translucent, so you can see it when you hold it up to the light.
Part 3 may be found here.