That was changed sometime, I think without Blizzard telling in the patch notes. Tried it out with my two accounts (yes yes, evil multiboxer), and it works like this:
If two characters tag a mob (not in a group):
- No one has skinning: Mob is FFA for skinning afterwards.
- One char has skinning: Mob is locked for skinning to that char. Players that have skinning, but did not tag the mob (did not participate in the kill) can TRY to skin the mob, but it does not yield any leather.
- Both chars have skinning: The first one to tag can skin that mob.
Bonus: The first character can skin the mob without the second character needing to loot the mob. The mob disappears after skinning, but is still visible to the other characters that can loot it. As soon as they loot it, it disappears for them, too.