The Carnegie Collection, like Schleich and the Invicta models from the British Natural History Museum, prided itself on producing high-quality, hand-painted models that put accuracy and educational value first. In its ads, Invicta boldly asserted that its dinosaur models were not toys, like the kinds of cheap plastic dinosaurs you might find at regular retailers. Instead, they were miniature museum displays to be prized by adult science enthusiasts. The NHM went so far as to limit which retailers could carry their Invicta line, basically allowing it only at museum shops and other high-end gift shops.
Safari took a more middle ground approach, doing more to market their Carnegie Collection to a wider selection of toy stores and, eventually, giving them colorful paint schemes that would appeal more to children (Invicta quickly followed suit, beginning to paint their own models after nearly two decades of plain, austere hard plastic).

Still, they did stress the authenticity of the models above all else, promoting the fact that each came with its own "diploma" - the parchment-paper info tag (these would be quickly replaced with cheaper plain brown paper tags by 1990, and, in 1996, with glossy, full-color tags).

According to sculptor Forest Rogers in a 2009 interview with Dan Liebman for Prehistoric Times magazine, the design process for the Carnegie line was mainly an in-house affair, with only the final production prototypes being sent to Safari for manufacturing. Paleontologists at the museum (initially Mary Dawson, later Matt Lamanna) would select new species to be made into models for the collection, and send the list along with reference materials to Rogers. (Secondhand reports suggest that Safari itself also had some say in which species were selected - there have been instances where the paleontologists at the Carnegie were opposed to certain species being made, but Safari insisted, e.g. Ichthyosaurus). Rogers would then sculpt a first draft of each model using polyclay over an armature. She would then send photos of the sculpt to the museum, which would often have them checked by other experts before giving initial approval. This process would be repeated again after any alterations to the sculpt had been completed. Rogers would then ship the final model to the museum for one last in-hand inspection, and it would then be sent along to a company in Rhode Island called Replicast, who would take the polymer clay sculpt and create latex molds from it. Using these, they would create several hard resin blanks. These were sent back to Rogers, who would paint some of them as color prototypes. Once the colors were approved, the casts would be sent to Safari. Safari would send them along with their other models for tooling at their Chinese factory.

Making Molds
To my knowledge, Safari has never gone into a huge amount of detail about its tooling and production process. Their early partner Schleich has done so on its web site, and in an episode of How It's Made, the processes are probably largely the same, especially for the early models which were made at Schleich's factory.

The first step in this process is making a master model. Rogers' wax sculpture would be immersed in liquid silica in order to create a silicone mold. When the silicone cured, it would be cut open to remove the wax original. Then, the resulting silicone mold would be filled with liquid hard plastic. These hard plastic masters are more durable and would stand up to the rest of the mold-making process much better than the wax.

One of the hard copies would be sent to tool makers to create production molds. The hard copy is buried halfway in soft putty, inside a metal box. The top half of the model is then covered in plaster. After the plaster sets, it is removed, and the hard copy and putty removed. This creates one side of a plaster mold, and the process is then repeated with the other half. These concave negative molds are then used to produce positive convex casts which, in turn, are covered with liquid metal, latex, or other durable material (sometimes the plaster itself serves as the production mold) in order to make the final injection molds. (It is into these that the stamp text is impressed). In some cases, for more complexly shaped models like Triceratops, more than just two halves might be made and assembled into the final mold - depending on the pose, the mold makers might choose to make the molds asymmetrically as well, which is why mold lines on the finished models rarely run exactly down the midpoint of a model.

The production molds are mounted in an injection molding machine, and filled with liquefied plastic appropriate to the model (harder or softer depending on how many thin pieces it had). This would be melted down from PVC pellets (containing dye from 1996 on) and injected into the mold. This would result in a PVC blank, that would be sent to the painting department.The machine cools the plastic down within seconds, and the plastic models can be pulled free from the molds by hand - however, depending on the kind of plastic used, the models still need to be immersed in a cold water bath to harden them further and prevent warping. The cooled model would then be sent off for painting.

Safari has released some videos of the painting process it uses at its factories today, which are probably broadly similar to the early days of the Carnegie Collection. Below, you can see a video of a cheetah model on the painting assembly line. According to a video summary of the line by Safari, each Carnegie Collection figure has over 20 paint applications. This statement was presumably referring to the color vinyl 2010s era, so older figures likely had even more.

For various reasons, molds were often retooled or copied at some point during production. Safari periodically performed minor retools, like scratching out old belly stamps and re-stamping them. Mold fatigue could also be an issue, where repeated injection began to warp or soften the details impressed into plaster molds (metal molds, while more durable, are also much more expensive - some companies opt to make cheaper plaster or latex molds, knowing they will need to be retooled more often). 

In these cases, the molds would be re-tooled, or new molds would be created, in order to freshen up the sharpness and detail, often resulting in differences from the original model. For example, many of the Carnegie models released during the first few years of the 1990s had noticeably deeper furrows and wrinkles, and and overall "chunkier" appearance, compared to their 1988-1990 counterparts. It appears as though at some point, the decision was made to copy the original figures, probably using a pantograph machine, in order to create a second set of molds. Forest Rogers has described this effort to "renew" the molds as a game of Telephone, where the pieces "gradually get further and further from the original sculpt." 

Whatever the reason for the Carnegie's frequent retools, in 1996, when production switched to the use of colored vinyl, the molds were more sharply retooled or at least partially re-sculpted (Forest Rogers was apparently not involved in this process). In some cases, new sculpts were made based directly on the old ones. For example, though the 1988 and 1996 Stegosaurus models are in the exact same pose, the texture and detailing is noticeably different, to the point that it seems the model may have been totally re-sculpted based closely on the original, rather than fully re-tooled. The Tyrannosaurus was a more major retool, again in the same pose but with all-new skin textures and a totally different head sculpt. Though many collectors don't count these as "re-sculpts", but rather re-tools, the difference is really just one of degree, because a retool always involves some level of re-sculpting - just not always by the original sculptor. What really matters is that as new molds are made from older production samples rather than fresh castings of the original resin prototypes, each mold differs to either a small or large degree from the version the sculptor originally created.