Yes I did mean paraguay, which has excellent pigeon shooting but good shooting can certainly be had in Argentina, depending on the reqion that you're in.
On the dove shooting, they are smaller birds, like our Mourning Doves here in the States and can be shot in a number of ways. The main shooting is pass shooting as the birds move from the roost to the fields and also they can be shot in the fields. With the large number of birds the outfitters move you between locations in the morning and afternoon shoots to give a variety of bird presentations.
Many who've gone to Argentina say they don't want to shoot more than 500 shells per day and then the birds start flying and suddently their shell count is at 1000 + for the day. It is very hard to hold back until you've shot a few thousand shells.
The pigeons are hunted more in your style, with field shooting, hides and decoys used, but they are also present for pass shooting as they fly with the doves. So, with a good bird boy it is possible to pick the pigeons out when they fly with the doves, as they are much larger than the doves and shoot them.
The outfitter puts a limit of 200 birds ( pigeons ) per shooter per day on the pigeons as the numbers are smaller and they do not reproduce at the rate that the doves do. With the doves they are millions of them and they consume 25% of the crop, so the land owners want them shot.
The distance from the States to Argentina is great also, but now is the time to experience the best wingshooting in the world so, if you're up for it you may want to consider going.