How much say does an associate professor have in accepting a foreign intern into his lab?
Different schools have different procedures, but as far as I know, a formal offer of admission will come from an admission officer (or a graduate officer), not from a professor.  

Usually, the most a professor can do is to recommend a student for admission to a program, and possibly indicate that she/he has already met the student.  The admission process itself is usually handled by a separate officer, which will check if all the admissions requirements are met.  Those might include various academic requirements, suitable background in the primary or a cognate field, language requirements etc.  (These requirements might be checked at the departmental level, and a recommendation then forwarded to the appropriate admission officer.)   

In some cases, admission may depend on the availability of funding, so that a student who applies knowing a professor can guarantee funding has an advantage. 

Funding is usually the trickiest part, but I am also aware of cases where a student had found a willing supervisor but the file was rejected on academic grounds.