Greetings from Nebraska.
Although plastic blanks can be successfully drilled with standard high speed steel drill bits (120-degree grind) and brad point bits I suppose, however I prefer using bits made specifically for drilling plastics. The bits are ground so they bite into metal as they are pushed into it. In plastic this action results in chips and more friction which causes heat. Bits that are made for plastic have more of a sharp point (60 to 90-degrees) which allows for a more gradual penetration (less bite) that reduces chipping and reduces friction (less heat). This is something I learned from managing a fabrication shop for many years that worked with plastics and ceramics as well as common metals. If you are interested, I get all of my drill bits for plastic from McMaster-Carr.
For successful drilling into plastics with standard bits, take it very slow, more like a slow pecking action (drill a little then pull the bit out) so that the bit can cool down and the flutes can cleaned out. Material packed into the flutes causes more friction (heat) and pressure inside the blank. Some people use a lubricating coolant when drilling plastics (water, oils, etc.), but it can leave a little bit of a mess on your drill press or lathe if you are drilling on the lathe.
I just saw your response that you are asking about a reconstituted stone blank. Although reconstituted stone blanks, FauStone, TruSonte, SimStone, etc., typically have a plastic binder, they are mostly powdered stone that has been compressed. For these I use regular high speed steel bits. I have heard that some people pre-drill using a carbide, masonry type of bit and then use a standard bit to get the right diameter. I have splurged for carbide drills from McMaster-Carr for a couple of the pen types that I use frequently (10mm and 3/8-inch). These bits are harder, stronger, and more wear resistant than high speed steel or cobalt steel bits so they retain a sharp, hard edge even on abrasive materials. They are however very brittle so they need to be used in a drill press, lathe, or mill i.e. no handheld drilling! The same in-and-out, pecking, cooling, process is used to make sure that the flutes don't get packed and to allow the bit to cool.
Since standard drills usually cause more problems (grabbing) when exiting a hole, another great trick is to cut your blanks long, measure how deep you need to drill, drill, then cut the blanks to length.
Regards,
Dave