Most flag states accept each others CoCs reciprocaly. In most cases these are obtained by attending full time schools/academies run by the Government and fully compliant with STCW standard curriculum w/latest amendments.
Here is the requirements to obtain CoE from MPA Singapore:
And a list of countries whose CoCs are accepted by MPA:
And the same for NMA, Norway:
You will notice that USA is on the MPA list, but not on the NMA list, although Norway has unilaterally approved US issued CoCs for service on NIS ships, but not for NOR registered ships.
Other STCW courses, such as Sea Survival, Fire fighting etc. conducted by private but approved institution, like Falck, MSTS etc. are also mutually accepted between flag states, but because there are some “fly by night” and less than serious operators that offer such courses, this is more restrictive.
For education and training institutions to get approval by NMA the following guidelines must be complied with:
You’ll notice that Nationality is not mentioned as a requirement, only inspection by NMA, or a Maritime Authority Norway have approved.