My dad and my grandpa each bought .44 Mag Desert Eagles back when the Mk VII was the new model. Since then, they've each taken several head of game with them, and enough rounds to have gotten several times their entertainment value from each of them. My brother and I have been shooting these pistols since our early teens. My brother thought enough of his early experiences with the pistol to buy a Mk XIX .50AE during his first deployment. That is worth its weight (significant) and cost in entertainment very quickly--mucho bigga fireball!
In general, Desert Eagles are well built, durable, and reliable. In our experience, they are accurate as rail guns and have no real special maintenance requirements, other than that you do clean them every once in a while and stick to jacketed ammunition. The .44 Mags are very tame. Even with the hottest loads you can stick in them, the .44 Mag Desert Eagle has been the softest shooting .44 I've ever had the pleasure of firing. The .50 is considerably more so, but you understand and expect this when you see the cartridge it fires. It still isn't unpleasant--more just 'wow.' The fireball is impressive.
Speaking of which, one time my grandpa double charged one of his handloads for his Desert Eagle. He said "it just felt weird going off." He had to have the molten brass cleaned out of the gas port and firing pin channel, but no parts had to be replaced, and the pistol still functions flawlessly.
They make decent hunting pistols, though it does suck losing brass in the brush, esp in the case of the .50AE and .440 Corbon. They are more than adequately accurate and powerful for this purpose. But they are undoubtedly more for fun and entertainment value than anything. The Desert Eagle provides a decent amount of this. It is a unique enough shooting experience that I think everyone should at least try one. If you're looking for that sort of excitement from an auto, the Desert Eagle is about the only game in town.
You do have the manhandle them as they can be prone to limpwrist failures, in our experience. The pistol has a large grip. It feels large even to those who have no problem handling other large handguns, like the full size Glocks and USPs. And while the Desert Eagle's weight does help control recoil, it can be ridiculous sometimes. It is a well machined five pound chunk of metal--end of the world durable--but very quickly heavy enough to be tiresome when held out in front of the body at arm's length. Find a good holster if you plan to pack it around.
It's nice having a handgun with a more bassy, throaty roar, larger fireball, and louder visual and auditory effect on target at 100 yards than the other guy's poodle shooter.