How do you hang something on sheetrock where there is a plastic stud?
How old is the building you are working in? Only a very new one might have (ecostud.com/product_details.html ) plastic framing studs. More likley you have drilled into a plastic pipe. If your water is on and you haven't noticed a leak you have drilled into a drain line or a plumbing vent stack. If it was my house I would cut a small opening in sheet rock at that very same location to verify what you have drilled into and if it is in fact a pipe open the wall enough to repair the hole in the pipe.

Please let us know what you find. 
And if it is a stud made of plastic explain why a screw would not pierce it.  Studs are after all supposed to be accepting of dry wall screws regardless of what they are made of.