Are there softeners used in polyester or polypropylene to make these materials more flexible? Or is this not needed for this kind of synthesis?
Yes. All plastics have plasticizers in them. The particular plasticizers chosen depends on what properties are desired of the plastic. Some can make the plastic much stronger but also more brittle at the same time, so that it would be much more difficult to break but when/if it did snap, it would tend to shatter or snap violently. Others make the plastic softer and less brittle, so that the plastic form has less structural strength, but will bend, stretch and disform under heavy load instead of snapping.
Plasticizers can also affect the temperature resistance to deformation, volatility (temperature resistance to deflagragation, and properties of said polymer if ignited, such as, will it go out on its own or will it continue to burn the whole thing). Plasticizers can be added to affect the polymer's UV resistance, resistance to chemicals or resistance to abrasion.
Once again, all plastics have plasticizers in them. If one were to magically remove all of the plasticizers from a plastic bottle (PETE or PE), the plastic would become brittle, dry, and probably reduce to tiny pieces (think shattered tempered glass, but smaller) or a powder upon touching the plastic form which no longer contained any plasticizers. In fact, you can observe this with some old plastics from the 1970s. Even though the plastic may have never been exposed to UV light, a plastic that is losing its plasticizer can take on the same cracked, opaque appearance. Most plasticizers are volatile at room temperature and will slowly evaporate away just by simply being exposed to air.
Most plastics will leach their plasticizers more rapidly after being washed with soap and water, after being microwaved, while being heated, or while at a low temperature (such as in the freezer). Even still, some plasticizers will take 20 years or more to evaporate away enough to change the properties. Some plastics will never lose their plasticizer (that we know of).
According to Wikipedia:
Some of the most common plasticizers and their uses are:
Phthalate-based plasticizers - used in situations where good resistance to water and oils is required. Examples include Bis(2-ethylhexyl) phthalate (DEHP) and Butyl benzyl phthalate (BBzP).
Adipate-based plasticizers - used for low-temperature performance or resistance to ultraviolet light. Some examples are Bis(2-ethylhexyl)adipate (DEHA) and Dimethyl adipate (DMAD)
Trimellitates - used in automobile interiors and other applications where low volatility and resistance to high temperature is desired.