Record Cleaning Mystery

Do you hear anything different from your cartridge?  If I were you I'd make sure your stylus isn't damaged before proceeding. I know this from painful experience. Once by sheer coincidence my stylus got damaged just after I bought a new record cleaning machine. Took me a while to realize that the filaments on my stylus were actually shreds of vinyl (and had nothing to do with the cleaning process). That was fun! I hope you don't have that problem but it's worth taking a look. (if I rubbed the filaments they turned into dust). 

If only on one side, I think you may be picking up dust or dirt somewhere during your cleaning process.  If you clean side one first and then side 2 and the fuzz only appears when you plan side one, I think you may be "contaminating" the clean side somehow when you flip it over to clean the other side. 

@swampwalker’s theory is as good as any. Here’s another: you’re record cleaning process has succeeded in loosening stuff on your records through fluids, brushing and you’ve vac dried them, but there’s still contaminants on the record. I’ve had this happen on various occasions-- the stylus gets fouled by a "cleaned" record that really isn’t fully clean.
Only solution is to hit it again, and maybe change up your method if you are finding this consistently. I’ve never used your machine, but it looks very much like a wand type (VPI)- if it has a slot in the vacuum wand, it should be positioned just so to maximize vacuum power and minimize sticking to the record. The velvet lips have to be kept scrupulously clean (you probably do that, but just for completeness) and of course, overvacuuming creates static. If my theory-- which I’m only throwing out as a possibility-- is the cause, you might also think about a pure water rinse (distilled is OK, there is better) with a different wand if you can do it, but my bet is, your cleaning is causing the problem if the stylus and platter mat(s) of both your turntable and cleaning machine are clean.
NB: Looked more closely at a picture of what I assume is your machine, there is no platter as such -more like a Nitty Gritty in holding the record, not fully the size of a 12" LP underneath, so that shouldn't be an issue. 

Thanks to all for your great replies. I will look at the lips on the vacuum arm. Maybe they need changing.  I do pick up dust on both sides of records and yes there is no platter on the Pro-Ject cleaner.  Obstruction is an interesting thought. The last time I went to empty the recovery tank hardly anything came out.!

Pure water rinse is always a must, in my opinion. I do double rinse. Okki Nokki machine, not yours. Play couple of uncleaned but not too dirty records or clean a few records manually and see what happens.
Checking the stylus condition was a really good idea.