I think that paper will be a luxury item. Most people will see it as an inefficient information storage medium. It will have more mass and volume that digital storage and mass = money if you have to move it. Extra mass needs extra reaction mass (which, itself needs extra reaction mass, etc.).
The mechanical problem of making paper is easily solved since the only part that requires gravity is the separation of the pulp and the water. I see two ways to overcome this. Spin the screen (as has been mentioned elsewhere) or spray at high speed onto a nonstick surface (make a paper "paint" that you peel off.
The real kicker that I see is that you need fibers to produce the paper. For efficiency, I think most hydroponic plants will be selected for their consumable to waste ratio. As such, most plants in space will have no or minimal fibrous parts. So, you would have to plan ahead and select non-optimal plants.
Another source of fiber is animal hair. I think that wool is best but I don't know how efficient sheep will be to raise in space. I've heard that goats are the most likely for small habitats.
This may not be too much of an issue if the clothing is fiber based. In that case, there is already a fiber producing infrastructure and paper can be made out of worn out clothes.
If I remember my James Burke correctly, monks raising sheep led to cheap wool underwear. Plentiful worn out underwear, turned into rags, led to cheap paper which led to the spread of literacy.
The same mechanic could work in space. You just have to figure out the most efficient way of producing the fiber in the first place.