I recently acquired an all steel pistol in 98% condition.. The box says, .32 caliber Puma Beretta.
There are two different pistols that Beretta has marketed in the USA as the Puma.
The first was a postwar version of the Model 1935, a somewhat squarish-looking pistol with a rotating thumb safety above the trigger guard.
The second, known as the "New Puma", was the much sleeker-looking Model 70, introduced in the late 1950's. This gun will have either a crossbolt safety at the rear of the grip (to mid-1968) or a 1911-style frame-mounted thumb safety (after mid-1968). However, AFAIK most Model 70's sold as Pumas were imported before 1968. Pre-68 pistols also won't have frame or slide markings referring to Garcia, Berben, or Beretta USA; post-1968 pistols have these markings because the frames had to be made in the USA to dodge import restrictions.
Lots more info here:
http://berettaforum.net/vb/forumdisplay.php?f=40
I wouldn't recommend monkeying with the trigger on these guns, because they're becoming collectible, and nice all-original examples are scarce. I generally prefer for people to stick to more easily replaceable project guns.
One other thing, if you haven't figured it out already...
If the slide slams shut by itself when the empty magazine is removed,
the gun is NOT broken. Most Model 70's and all Model 1935's lack a mechanism to automatically hold the slide open when the mag is removed. The shooter is supposed to lock the slide back manually before removing the mag. On the Model 1935, this is done with the manual safety, whereas the Model 70 has a dedicated slide stop lever.