Unable to comment on other operator nations in the world, however with respect to the UK Chinook force - yes most aircraft carry the ability to have the hoist fitted, in my own experience the crews employed were not SAR current and had no reason to remain so being part of SHF and not the SAR Force; the crews did however practice hoisting from minefields (or similar hostile environment) before deployment in support of troops.
Chinook & Puma crews who were in direct support of SF operations were (and possibly still are) required to maintain a winch currency, and indeed most SF support airframes carried the winch in my day - although not the Puma due to the space it takes up in the cabin when stowed.
One amusing event from my days as a spanner on the Puma force, was seeing a baby loadmaster dragged across RAF Abingdon airfield grass by a hovering Puma who were doing not a very good job of holding a stable hover - although my smiles were short lived, as it was my turn in the strop next; luckily for me the crew plucked me out of the long grass far more professionally than the poor fellow before me!