I'm a New-grad RN from Colombia and of course studied my major in Spanish. Should I put the names of the hopitals I did my practices in, in Spanish or English, or both?
I would do it the same way I expand abbreviations by putting it in brackets, for example,.
NUS (National University of Samoa)
So if it had two recognised names or even just for clarity.
The National University of Samoa (Le Iunivesite Aoao o Samoa) or vice versa if the primary language was Samoan.
This is the conventional method, so best to use since everyone understands it.
Disclaimer: I did not study at the NUS it's just an example, and I in no way imply that hiring someone from there is a good idea.
Some institutions have also an international title (the one I graduated from is called "Athens University of Economics and Business") in that case use that. If the institution has only a Spanish name use that. Expecting the reader not to know it provide a short description on what it is and what it does.
My take from my CV and those of friends that I have edited is to use the name with which someone can do an Internet search and locate the institution. Do not forget to add a link to institution's site if this exists (again Eng. version of the site if applicable, local otherwise).