New York medical schools and their competitors offshore are clashing over a precious resource: the opportunity for students to watch and learn from doctors in hospitals. Students in their third and fourth year of medical school need clinical clerkships so they can see how doctors diagnose patients, perform surgeries and deliver babies. Officials at New York medical schools say it is becoming harder to find sites for these rotations, partly because for-profit offshore medical schools, often in the Caribbean, have bought slots from New York hospitals at prices topping $400 a week per student.