I suspect that there is not one answer to this question as paper was used over a long period of time and would have gone through several changes. Evidence for this is paper thin (pun intended) and lot just circle back to the mythbusters episode, but I have managed to find a couple of sources that give some indication of how it might have been done. From this I have two methods that seem to be confirmed via historic sources and I have an unsupported idea from my own supposition.
The first of these is waterproofing using lacquer (a la JBH's answer) and resin. One of the things I found was that Korean paper armour was considered to be particularly fine, and was a common import/tribute, so looking for Korean Paper Armour I managed to find this:
http://cejsh.icm.edu.pl/cejsh/element/bwmeta1.element.desklight-9260b5e1-b88f-4bf8-9521-9cd0ea330675/c/randall.pdf (Note link downloads a PDF)
It references a report on the materials required to make 120 sets of paper armour and works out there is roughly five litres of varnish and one of lacquer (which lines up with what what JBH has suggested). In the same article though it notes that even with this mould and mildew was a persistent problem that was inspected for.
The second is cloth. In this source:
There are a lot of references to the armour being constructed of alternating layers of of cloth and paper, essentially creating a gambeson with paper additions. In this situation the waterproofing, or more likely stabilisation of the paper when wet, would have been provided by the cloth layers, potentially with whatever waterproofing methods were using on cloth at the time.
Of these it looks like cloth settled on being the preferred option as the sources indicate that is was still around in the 19th century to be tested against western firearms. However this might not indicate effectiveness of waterproofing (or defence) rather good enough performance combined with ease of manufacture.
My own supposition is that are that there were designs that were inherently waterproof, though from different methods. This is my own supposition, from a couple of unsupported lines I read while finding the sources I have cited, so take these ideas with a grain of salt.
The first of the two was that paper armour was something akin to Geek linothorax (https://en.wikipedia.org/wiki/Linothorax), essentially impregnated with glue before being layers upon layers being being stuck together and finally beaten to restore the paper's flexibility. In this method the glue becomes the waterproofing.
The other one is that we are thinking of the wrong sort of paper. A couple of places on the net indicate that barkcloth, a material made from beating parts of bark into sheets. Others, however, reject this as it as primary sources use different terms for paper and backcloth (https://greatmingmilitary.blogspot.com/2015/02/paper-armour-of-ming-dynasty.html). Even if it was true paper that was used we could be looking at a type of cloth based paper like that used in US currency which holds up fairly well to being wet, although it's not great for it.
If these were used I doubt they hung around. The glue type would have required a lot of materials and been time consuming to make, and would have likely been out performed by metal. The cloth type would have probably been prohibitively expensive as the best cloth base would have been silk which would have been expensive and if you have silk you might as well make a silk gambeson instead.