Depends what it is and resultant condition of the impact; often not worth the effort! Yes I do pick them up if I see them hit and it is practical to do so. Venison a particular treat if truly fresh but very rare to get it.
tried a bit of deer once and not keen, also tried pheasant and that's got an odd taste as well, used to go shooting many moons ago with some mates and bagged pigeons, rabbits and pheasants, all to eat not for sport, thinking back seems really odd now to me, i do remember pigeon meat was ok, couldn't shoot a living thing now, funny how differently you view things as you get older, still got the 12 bore Berettas but just do the occasional clays