I have 2 Beretta Cougars and one Stoeger Cougar, all of which have a rotary barrel breech lockup mechanism like that of the Beretta PX4 Storm full-size and compact.
My opinion is that the rotary barrel mechanism is happier if it is run a bit "wet". I think this is the general consensus judging from what I have read on berettaforum dot net and other forums over the course of quite a few years. But there are dissenting opinions.
I use a thin film of grease on the bearing surfaces within the slide that the barrel rotates on, on the barrel itself, on the camming grooves on the barrel underside, on the central block, and especially on the tooth of the central block.
If you field strip, clean, and lubricate your pistol frequently, oil will probably do just as well.
When my Cougars are starting to get a bit dry, I can feel the action starting to get balky when cycling the slide by hand.