A nice .357 makes a great small to medium sized critter getter.Load a revolver with some cast lead,super accurate fodder and as the ole timers used to say....."you can eat right up to the bullet hole".IOWs,head shots on squirrels and rabbits is entirely doable.
Theres some "issues" with .357 once the animal size gets above Fox,G-hogs,Turkeys,ect.......Thinking Deer and above.One issue is of revolver strength.To get decent ballistics heavy bullets and large doses of powder can take its tole on a revolver.Large framed guns like S&W # 27/28,and Colt Pythons will last awhile....but its gonna wear them out faster.
Another issue with .357's,if you aren't casting your own is,bullet availibilty.Most of the industry sees the .357 as a defense calibre....and small game.Bullet choices can get a little slim when looking for say a "Deer" bullet.But thats revolvers,TC's come in .357 and can make that cal really stand-up and perform.But the .357 maximum,IMO would still be better choice in a TC.
.44's,when loaded correctly make dandy short to medium range deer guns.Handgun hunting is IMO,probably the toughest hunting discipline.I think its noticibly harder than bowhunting.It takes a tremendous amt of practice and load development to really become profficient with a handgun.Scopes help a little....shooting sticks help a little/lot.....its like,everything helps a little.Its just much harder than say a rifle.And thats why I see it mainly as a short to medium range gun.
Theres a much better bullet selection for hunting with the .44.You're not having to go into the red zone(like a .357)to get the ballistics needed for quick kills on Deer.So finding suitable guns isn't that big of deal.For instance,a scoped Super Blackhawk shooting factory loads is one heck of a pkg.And won't break the bank.
I will say,hunting big game with a handgun....and being successful,ranks about as high as it gets on the satisfaction scale,for me at least.Goodluck,Shopnut