Ulil Amri , ruler is the one responsible for doing so.
I ask a simple question, assuming France is not an Islamic country altogether, what if a criminal commits a punishable crime (in an Islamic community) like murder, rape, adultery etc. then supposing Some_Country is a completely Islamic country altogether, can these people Some_Country take actions going on in France? Can they punish the criminals in France just because Islam has laws provided for those crimes?
The answer is basically NO.
In a community say A, a leader of some other community say B doesn't get to take actions on the people of A. So, in general, only a leader of an Islamic community can take actions only in his community.
Now in our case, if Muslims are a minority in a community say A, then Muslims have to be very careful in that place, abiding by the rules and regulations of that community. This could be considered analogous to Prophet Muhammad (pbuh) dwelling in his community of disbelievers during his early days of Prophet-hood. Imagine, at the very beginning when he was only with a few followers, when he was mocked. Only when a community comes in the hands of Muslims, then the leader of that community gets to rule according to Islamic laws and take actions as deemed necessary for the sake of the community.
So, if in a Muslim community (today), suppose some nonbeliever in it mocks the Prophet (pbuh) by any means, say cartoons, then the leader of that community decides what to do.
Here are some of the main points from IslamQA article which I found relevant,
We must denounce it forcefully, each of us according to his abilities, by sending letters and articles, or making phone calls, to their government, foreign affairs departments and newspapers.
We should demand a clear and sincere apology from them, not deception or justification of the crime which they call an apology. We do not want an apology for showing disrespect towards the Muslims; rather we want an admission of error and an apology for that error.
We should request them to punish the criminals for their crime.
And so are other points in it.
But, remember, anytime, what ever step is taken, it is taken wisely, rather than in a rage.
There's no Islamic law against those who insult/mock the prophet, the executions that took place in the past was during war time.
Today, most people refer to this Hadith when talking about execution of those who insult/mock the prophet:
Volume 5, Book 59, Number 369
In summary, the Hadith refers to Mohammad's (pbuh) orders to execute a Jewish poet who incited people against Muslim and Islam.
This isn't just an Islamic practice, even America commits such executions, for example the execution of Anwar al Awlaki, who was inciting people against America with his eloquent speech.
As for who can carry the execution of those who incite violence against Muslim? Anyone, but the orders must come from Muslim leader.
NOTE - this is not for mocking or insulting the prophet, this is for inciting hatred and violence against Muslim and Islam.