How do you know when one of the medicinal plants has been mastered?
I've been researching them for ages and still haven't mastered them all.

Are you talking about mastering your medical ?
If you mean learning them so you can get your herbs puzzle finished. That's really easy.
When your villager walks away, and doesn't shake thier heads no.you have mastered that plant.
Then You do the same thing with all the plants.You drop your villager near the plant.It really isn't all that hard.

Are you sure you're getting all of them? There are 4 plants. You can find more information about the puzzles in the Virtual Villagers Guide.

It's the lilly, the cactus, the rose and the plant that looks kinda like a cabbage(however unlike the others it doesn't seem to have a name) I gather? Well I've tried getting the villagers to study all of them and a master doctor has even done it, he doesn't shake his head either. So what is going on?

You should see a message at the bottom as each plant is mastered. The puzzle screen and completion message are triggered by mastering all 4. It's easy to miss one.

So how around how many times does each plant need to be study before it's mastered because I've been researching them for ages and does it have to be done by the same person or is cumulative?

Each plant should only need to be studied once, are you sure you haven't already mastered the puzzle, because usually when my villages are studying the plants, they only need to go on it once, and they never shake their heads.

The only reason I can think of is that it's some kind of glitch because I haven't solved the puzzle.

I agree with Izzy. Once around to all the plants should suffice. One puzzle down!

Well either way it didn't work in the first game I had going and I did all that stuff, I still think it might have been a glitch. I started a new game and tried it and it worked so I'm happy about that.