The style that is best depends very strongly on the intended effect on the intended audience.
The context might also include such things as who is doing it and what form "requiring" and "mandatory" were to take. It could be a law, a social expectation, a guild rule, or any of several other things.
Note also that there is a very small difference in meaning between the two forms suggested, and that will highlight the difference in effect they might have.
The "requiring" form emphasizes the action on the skilled persons. That is, the thing that is having the effect is the heavy-hand requiring of those people that they do something. The action is aimed quite directly at the skilled persons. You might have better effect using this form if you were speaking to a group of people who were short on doctors because many doctors had moved to another country. "We will require them to stay." Meaning that those who need doctors should reach out and prevent the doctors leaving.
The "making it mandatory" form emphasizes the action of creating a rule. That is, the thing that is having the effect is creating the rule. The action is less direct. You might use this form when speaking to a group of lawmakers such as the representatives in the country's legislature. "You should make it mandatory for them to stay." Meaning the lawmakers should make a law.
So either might be preferred, depending on the audience.
I will curtail my discussion of the actual advisability of any such efforts due to the effects on the morale of the doctors (or other skilled persons) so retained.