In France, that's what the master's degree is for. More precisely, the second year of master (colloquially known as M2) with the specialization as research master degree, as opposed to teaching master degree or professional master degree.
Advanced courses are taught during the M2. There is a huge gap between the level of the licence (bachelor) and the first year of master (M1), and another huge gap between M1 and M2. Students are expected to work a lot on their own. Moreover, in every math M2 program, the student is expect to do a 3-4 months "internship" with a professor and write a "memoir" at the end, a mini PhD thesis. This is very often where you get to learn the very advanced, very specialized material necessary to start doing PhD level research.
I should now state that there is a very clear difference in organization between Paris and the rest of France. France is extremely centralized, and this shows in math too. There is very large supply of M2 students in Paris, with people coming from Grandes Écoles or abroad just for the M2; meanwhile, outside Paris, M2 programs usually barely manage to get by with 5-10 students a year. I've experienced both (as student and faculty) so I can comment on what I know.
How this problem of regular teaching of many advanced courses on a certain subject is resolved on your math department?
In Paris, there is a wide variety of M2 courses on most subjects. There are enough students and faculty for this.
Outside Paris, not all courses can be taught every year, as there aren't enough students and faculty: some classes would be empty, some would lack an instructor (thanks to massification of higher education, we have to teach a shit-ton of classes to 1st year undergrads). A usual deal is to do thematic years, for example "this year we'll do a number theory program and a stochastic equations program. Next year, algebraic topology and dynamics." If a student wants to study something not offered, they unfortunately have to move.
This difference is also magnified by the rarity of PhD stipends. What good is giving someone a research master's degree in math, a degree that only prepares you to become a professional mathematician, to someone who has no chance of getting a PhD stipend afterwards? As you can guess, some universities are richer than others and can offer more stipends.
Who teaches all of them?
Faculty…? Who else? I don't understand the question.
Does there always exist a group in your area on your department so that the teaching load can be shared?
In Paris, yes. Outside, no. As I said, we have to teach a ton of classes to undergrads. And somehow, no one in the government was able to divine that babies born in 2000 would be college students today, so they haven't had the foresight to increase budgets for faculty jobs. So we are stuck with a dilemma: enrolling students in undergrad is something that we have to do by law, the number of students increases every year, and the budget stays constant. Guess what part of higher education is slashed first?
How it is decided who teaches what?
There cannot be a single answer. Every department in France has statutes and bylaws that can be (democratically!) changed, and this includes how teaching loads are divided. Still, every university I've been to has had this model: the department director delegates the power to either a committee or even a single person depending on the size of the department and customs. This committee/person deliberates on what courses can be offered and who teaches what. It's often (but not always) taking into account the wishes of every member of the department. Large department (= Paris…) have even more complex procedures.
PS: I hope this does not come off as an answer from a Parisian snob. The math departments in hard science-intensive universities in Paris are about 2-3 times as large as the largest math department outside Paris; moreover, it's only a metro ride away to the other Parisian departments, who can thus coordinate among themselves to have a unified program with much wider breadth. I have many colleagues outside Paris who despair at the state of their M2 program (and this won't be helped by the current government's policies of creating monster universities on the one hand and defunding smaller ones on the other hand… if you know how to solve this problem, please tell us).
PPS: This answer is mainly about pure math. The situation is less bleak in applied math. ("What good is pure math for anyway? Pure mathematicians can't get a contract with Thales or Safran to co-supervise a PhD student who will help us sell bombs to Saudi Arabia." --some government official, probably.) Some smaller math departments are slowly transitioning to applied math only.