You have to be really careful with the bedding thing. Modern shredders will happily accept staples which end up as sharp bits in the bedding. If the shredder will also shred cd's and credit cards then paper shredding may be contaminated even though the bin is emptied between shreds. There is also the issue of ink which can be dangerous for animals who pouch or eat their bedding. Personally I wouldn't be happy with a vet who used it unless they were SURE it was metal and plastic free.
If I saw an offer of shredded paper on Freecycle/Freegle, I'd take it for my open fire. Unless the paper is in tiny bits, it is useful for spreading under the kindling to get the log-fire going, and saves me screwing up old newspapers.
as atalanta says it depends whether it is in strips or tiny chips. If I saw it on freecycle and it was local enough I would compost it myself. How about ask your local allotments if they would take it.