I'm placing myself as Intermediate here (but only in the sense of the jump back not Ashtanga as a whole) in that I can VERY occasionally manage to get through without touching. Mostly my feet are brushing the top of the mat. This is a problem with a rug because I found I was just wearing my foot away, rubbing nasty holes in my feet just above the little toe where it joins the foot. This is also a problem with my jute textured mat (second from left in picture). This was my first mat and my favourite when I was doing beginner variations. I'm now using An extra long open cell mat (far left) the extra length is great. I have to be careful though, for the first couple of jump backs as I can rub the same place on my toe and can get a blister, same as with the rug. But as the mat becomes a little slippery from sweat my feet slide through and it feels like the real thing and I can focus on the other elements. I'm starting to think a black closed cell Manduka mat would be even better as there's no texture and it's probably a little more slippery. You can wipe the areas your hands and toes go with a towel to avoid slipping but keep it slippery in the middle where your feet go through. As I become more consistent in avoiding brushing the floor I would probably go with a rug over the top of the mat.
Advice for beginners? I'd say stick with a basic sticky mat, As you improve you'll find the sweat helps your feet slide through as you start to advance form the beginner variations to full jump backs.
WARNING about jump throughs. Trying to learn to jump though with straight legs can be dangerous with a sticky mat. A bunch of times I caught the sticky mat with my little toe going through and almost broke it. I highly recommend the bent leg Jump through via lolasana as a MUCH safer bet
PS. The light blue rug in the picture has a sticky coating on the back of the rug. This is a great idea as the problem with rugs is that they can move about a bit. I found the sticky rug a problem though. It's TOO sticky! put it on a wood floor and when you try to take it up it will take the varnish with it. Put it on a mat and it will tear off the top layer of the mat. Probably only good if you have a carpet but then, in that case, your probably fine with a normal rug anyway.