I take care of some rentals and in the course of time I've cut up and hauled off a ton of carpeting of all kinds. What I use is a utility knife, and replace the blades as they dull.

Get a slit cut in the rug for a start so you can get your fingers under it (Careful! Fingers and knife are not friendly), even if you have to kind of rub a hole in it, then aim directly at the backing.

Jute backing is the easiest. Once you get enough of the carpeting detached, turn it so the backing is on the outside and attack it from there. Jute is made of strands- cut those strands and it just melts down.

It wouldn't hurt to wear a paper mask while you're doing it- I've never had to do a sewage contaminated carpet but I can imagine that on the underside of the carpet you're going to run into some sludgy stuff...

Same with gloves, and washing hands frequently with soapy water, couldn't hurt. Any cut or abrasion will be exposed to megagerms.