As with any subject, there are a broad range of people teaching jiu-jitsu and it's often discussed who should teach and what qualifies them to do so. Well, you know the deal, here's my thoughts...
It should go without saying that to teach somebody should have approval from their instructor. This ensures their instructor believes them to be capable of doing so and also legitimises the teaching as being part of their lineage. It also means that the instructor should still be training as a student and continuing their own training.
The most obvious question about teaching qualification is what belt level someone should be. Except for in certain circumstances (no higher level guys teaching in the area) there is no need for white or blue belts to be teaching (and if they are teaching, they should have regular contact with a black belt instructor). Yes, they might be excellent teachers but they will not have enough technical knowledge. An experienced purple belt and any brown/black belt should definitely have the required technical knowledge, but this is still no guarantee of teaching ability. The environment someone is teaching in also matters here; a black belt can certainly run a large academy themselves... a brown or purple might be able to, and they could definitely run a smaller affiliate. Lower level belts are also much better placed to be teaching if they are doing it at an academy run by a black belt.
Judging teaching skills is a tricky thing to do. People learn in different ways and someone might be good at teaching one sort of person but not another. There is also always a human factor, as an instructor you don't just show jiu-jitsu techniques but also have to deal with the various personalities and attitudes of students. You have to make sure the class as a whole works well; people actually drilling, avoiding subdivision into groups/cliques and various other things. Some people are definitely bad teachers; they don't plan lessons at all, they don't have a clear idea of what to teach, they don't teach connected techniques, they favour certain students... and many other potential problems.
I think the best way to learn to teach jiu-jitsu is to learn from someone good at teaching it. People shouldn't just decide to start teaching, they should spend time studying a good instructor teaching classes and ask them questions about how and why they do things. The perfect scenario for someone to start teaching is that they train with their instructor until black belt and then open an affiliate academy, but there will definitely be times when people start teaching sooner (in a country without much jiu-jitsu). The worst option for someone to start teaching is the desperate-to-be-an-instructor types who leave their instructor at white or blue belt and start their own club. These people will use any type of excuse for it... they want to have a "different philosophy" within their club or they "see themselves more as an instructor than a student", maybe they see it as a fast track method to get a promotion via a long-distance affiliate relationship or they are just the sort who rate themselves much higher than their actual skill level. The worst of them will develop their "own style" and wear a black belt... whatever it is, they are idiots.
There are no official qualifications to teach jiu-jitsu, so it's always going to be the case that anyone can go out and set up an academy, but there will also always be people willing to tell others they shouldn't be teaching... and I'm one of them, haha.