Depends what you call a type rating.
Am I right in saying that, currently, an F/O on a 737/A320 flying domestic US, does not have to have a type rating?
Yes
Am I right in saying that, currently, an F/O on a 737/A320 flying domestic US, may not have practiced engine out flying in the sim?
No. They will have almost all the training the Captain had. They will have engine failures on takeoff both before and after V1. They will shoot ILSs with an engine out and land and go around with one engine inop. They won't do steep turns, they won't have a second engine failure on a three or four engine plane and they won't do a zero flap landing.
Am I right in saying that, currently, an F/O on an international flight, may not have a type rating?
To keep foreign CAAs happy, the FAA created an SIC type rating. The F/O has their Chief Pilot or training department certify they passed their FAA approved F/O training program and takes the paperwork to the FAA. The FAA will issue a type rating limited to SIC only without any further testing.
That being said, many international operators get their F/Os full type ratings since they operate long flights with one Captain and two or three F/Os and someone has to hold a full type while the Captain is sleeping.