I usually start off by trimming off as much of the dried flesh as I can with a scalpel blade. Got to be careful not to cut into the claw though. Once I get them trimmed as far as I can with the scalpel I use a 180 grit and sander bit on dremal tool to remove more of the flesh. You also have to be careful that you dont burn a hole through the claw when using the sanding bit.
Once all the flesh is removed I use the 240 grit sanding bit to touch up the claw. If there is any tiny amount of flesh remaining or where it has, splits, hang nails and rough areas.
When you work on claws that have been sitting out raw to long or have been tanned they can sometimes come apart. Bear Claws are not solid. They have a sheeth (claw) and a core (bone). Just like the hair on a hide that has been mishandles, claws can slip.