Hi!
Nothing to do with the carrier. It is the home-country's CAA (like FAA). You will get a license conversion, and have their home country CAA license. The type rating will be on that license. Usually, the conversion is not too painful. Sometimes, like with the Kenya CAA, it is a MESS!
When you get back to the states it is typically easy to get the type rating converted to your FAA license. If you have a lot of problems at a FSDO/with an FAA rep, go to another rep or another FSDO. Some places are used to converting, some aren't.
ALSO, many type ratings are different from US-overseas.
For example, a DC-9 type FAA is -9/MD-80/90/B-717. Most CAAs count them as 4 different type ratings. Another is the ATR. FAA is ATR. Vietnam CAA have ATR 42 and 72 as two seperate type ratings. Same goes for some of the CAAs and some of the CRJ/ERJ type ratings.
cliff