The College of Veterinary Medicine at Washington State University, provides the following information for Poinsettia, as well as Mistletoe and Holly. "Holiday Health Hazards" (https://hospital.vetmed.wsu.edu/2021/11/23/holiday-health-hazards-for-pets/)

"Poinsettias fill homes with color during the holidays. Poinsettias have received bad publicity in the past whereas in fact, poinsettias are not very toxic to pets. They do contain a milky sap that can irritate the mouth but if signs develop they are usually mild.

"Mistletoe can be very toxic to animals and you should seek veterinary consultation immediately if your pet has potentially ingested any part of the plant. Mistletoe can cause vomiting, severe diarrhea, difficult breathing, shock and death within hours of ingestion.

"There are many species of Holly (genus Ilex). Berries and leaves can be a problem although signs of poisonings are generally mild, and include vomiting, belly pain, and diarrhea."

The Colorado State Extension site has an extensive review of poinsettia history and cultural requirements at: https://extension.colostate.edu/topic-areas/yard-garden/poinsettias-7-412/

They include a section that discusses research indicating that poinsettia is not toxic but if eaten may cause discomfort. Those sensitive to the natural latex in the plant should, of course, avoid contact with it.

Regarding pets:

"Although commonly assumed to be poisonous to animals, Poinsettia plants are not harmful to household pets unless the leaves and bracts are eaten in very large quantities. Some cats that chew on the leaves may salivate and can vomit if the leaves are swallowed. Since cats and puppies frequently chew on new plants introduced to the home, it is prudent to place the plants out of reach!"

Additional articles on holiday plants from Purdue: 

https://www.purdue.edu/hla/sites/yardandgarden/how-to-care-for-poinsettia/

https://www.purdue.edu/hla/sites/yardandgarden/the-mystery-of-mistletoe/

More in-depth information on poisonous plants:

https://www.purdue.edu/hla/sites/yardandgarden/some-plants-are-poisonous/

https://mdc.itap.purdue.edu/item.asp?itemID=15946