Are you wanting to know if you should use paint, linoleum, etc on top of the plywood or are you asking about bedding materiel (straw, shavings, sand, etc.) that can be used - or maybe both??
My floor is not covered with any sort of additional product - I bed directly on the wood with a deep litter of pine shavings. It's really a matter of personal preference with actual plywood. An additional barrier such as paint or linoleum can help to extend the life of the plywood by providing an extra layer of protection from the wetness of direct contact with bedding and waste, etc., but it is not necessary. For any type of wood product such as a press board, OSB, etc that you might be using in place of actual plywood, that barrier would be more important.
I used some fiberglass / plastic shower board from Lowe's...it is fairly expensive at $30 / 4' X 8' sheet though...it has a raised pebble type of surface providing some grip for the chickens but it is still fairly slick. I used treated ply-wood that I aged outside under the awning for a couple of months but I still wanted more protection to prevent the floor from rotting. Our coop is 8X8 and I used to sheets of the glass board and sealed the seam with silicone caulk. I looked into Linoleum and you can probably get enough from an installer for free or very little that is left over from a job but it just seemed to slick to me. But I have no real world experience with it.
On the floor most use Pine shavings or sand as litter...good luck!
It has been down in my coop for 5yrs, and it looks like it did when I first put it down. Linoleum and vinyl will eventually wear and break down. Then moisture, and possibly insect pests can get under there and cause problems. I read on this forum a couple of years ago, where a guy put down linoleum. In a couple of days, his chickens had shredded it to pieces. The roofcoat will seal all floor seams and gaps, along with the wall to floor seam/gap. Leaving the wood floor unprotected is not something I would want to try on a long term basis. Especially if you let your litter (Pine shavings) build up, like I do. It can get a bit damp at the bottom of the heap under the roosts. That moisture will get to an unprotected floor eventually. Application is easy, just dump it on the floor and spread it around to where you want it. The most important thing with this stuff, STIR WELL.
here's the trick to keeping it down, go to Lowes and get pallets for free, take those and rip em 1 inch strips, take those strips and put on perimeter use 18 ga narrow crown staples and drive into floor. the linoleum wont come up.