I did for a while last year. Glued a bunch of 2" strips to some flat pieces of wood about 12" long and went at it. I use it like a big freehand Lanksy, knife on edge of workbench, eyeball angle and grind the bevel back and forth by hand. Turned out I can be slightly more consistent than freehand on a stone - which kinda sucks because I've been doing the latter for decades. Sharpened mainly low/mid-tier steels - H1, D2/K110/DC53, VG10, BDN1, LC200N - and some S30V and ZDP189. Using 80, 120, 200, 320, 600, 1200, and 2000 grits.
My comments are below, some are specific to how I set-up my sandpaper "system" but some are general to the topic.
Pros:
New sandpaper cuts very fast, it's way faster to reprofile a thick, obtuse edge using sandpaper than the stones or plates I have.
Set-up is clean and transportation is easy - no oil, minimal water, means I can throw some sandpaper sticks in my travel bag and have a decent sharpening system for the road of on a jobsite.
Seems cheap if you have tools or access to pieces of wood already.
Whatever finishing grit I used seemed to knock off the burr a little slower than on a stone or diamond plate. This made it feel a little easier getting the finicky knives up to sharpness. Cliff Stamp-style apex sharpening was the same way, for some reason it was easier to see the apex forming and stop before a burr was created. Maybe due to using the sandpaper dry or with light water only vs. oil on my stones.
By gluing the sandpaper on top of 1-2 sheets of notebook paper, glued together to the stick, it is possible to make the easiest convex-edge system I've ever seen. So if you like convex edges, sandpaper is the way to go. I never messed with convex edges until now and for my chef's knives, I'll never go back.
Cons
Lower grits wear out quickly, reprofiling a 3.5" D2 blade from 25° to 15° wore through two strips of sandpaper, so the time saved by how fast the stuff cuts when new, disappeared when I had to remove and glue on new pieces or cut new sections of wood.
Not actually cheap, I've been using the same three Norton stones, flattening stone, and stick of green chromox, for about two decades. They'll all easily last another two. My sandpaper system, if I sharpened at the same volume is at least a $40-50 a year investment, including about 3-4 times the time for maintenance of the set-up.
If you don't have a fine edge table saw blade, it's hard to find pieces of wood that are true and flat enough to work well. Any irregularity in the surface of the wood is magnified at the surface level of the sandpaper so a small ripple you can't really see can eat a chunk out of the edge quickly if you're not paying attention.
Low grits are no joke, the scratches that come from using 80 or 120 grit sandpaper are crazy deep, much deeper than the comparable grit in a stone or on diamond. You can see in the picture below, The bottom edge I had some issues removing them because my progression was 120->320->1200 since that was the progression I was used to with stones. For sandpaper I found I needed to add 200 and 600 to remove the scratches left from the lower grits. Doesn't seem to affect the cutting ability or edge holding but is aesthetically ugly. Uh...I'm going to fix it eventually.
All that said, I keep sandpaper around as another tool in the sharpening arsenal. It's good for some stuff, and I really like how easy it is to convex with it. I much prefer reprofiling with it since it's so fast, it makes for a more consistent edge than using a stone where fatigue/boredom can be an issue. Hope this was helpful.