As far as I know, it is no longer possible to obtain the original, shorter version of the novel except from used bookstores. All printings of all editions since 1990 have been of the longer version.
The original novel version and the original magazine version were definitely not the same - the magazine editor made unauthorized changes, and Heinlein did not like that at all. This is covered in his correspondence with his agent at the time, in Grumbles from the grave.
The uncut version of Puppet Masters did not get a new hardcover edition like Stranger did. Instead, around 1990, a new printing of the mass market paperback edition in print at the time (del rey I think?) switched from the shorter to the longer text from one printing to the next. There was no indication of this change anywhere except for a tiny notice on the copyright page and an "original uncut" blurb on the cover. The edition I remember buying around 1992 because it proclaimed it was "the original uncut novel" had this cover art:
Since the "original uncut" blurb soon gave way to a "now a major motion picture" blurb (and more recent printings may not say anything at all), here are two ways to tell which version you have.
- On the copyright page, the longer version may say that it is the "first revised edition, Jan 1990."
- On the first page of the first chapter, if the narrator wakes up in bed alone, it is the original cut version. If he wakes up with a nameless blonde, it's the uncut version.