Betsy, who is most likely named after a nursery rhyme, is introduced in Tik-Tok of Oz, which is sort of a retelling of Ozma of Oz with some new elements added. See, Baum wrote a stage version of Ozma, which eventually morphed into the play called The Tik-Tok Man of Oz. Perhaps because Dorothy had already been a major character in an Oz play, or because she had come to live in Oz within the book continuity by the time the play opened, Baum replaced her with a new Midwestern girl who arrives on the Ozian continent after a shipwreck in the company of an animal. Betsy is from Oklahoma instead of Kansas, however, and comes to shore with a mule named Hank instead of a yellow hen. And while Betsy is likeable and spunky (I always liked the scene where she sasses the Nome King), she never really takes center stage like Dorothy does. In both Tik-Tok and Hungry Tiger, she largely stays in the background and allows other characters to drive the plot. We never learn all that much about her background, either. Greg Hunter's story "Betsy Bobbin of Oz" gives the plausible explanation that her parents were blown away by a tornado, and she was on her way to live with relatives in England when the ship was caught in the storm. Also left pretty vague is Betsy's appearance, but that's pretty typical of Baum's child characters. Dorothy's blonde hair was an invention of Neill's. I generally think of Betsy as a redhead, as Neill sometimes drew her (probably largely to make her distinct from Dorothy), but a fair number of pictures make her a blonde as well. I believe it's been confirmed that, at least in Tik-Tok, Neill used Philadelphia-born actress Eleanor Boardman as his model for Betsy.
Trot is a little more developed, although a lot of that development occurred outside the Oz series. After Baum tried ending the Oz series, his next attempt at a children's fantasy series starred a girl named Mayre Griffiths, nicknamed "Trot" (and sometimes further nicknamed to "Tiny Trot," due to her size). Her father is a ship captain who's usually out at sea, and thus suggested that his seafaring friend Bill Weedles (better known as Cap'n Bill), who lost a leg at around the same time as Trot's birth, stay at his house in California to keep his wife and daughter company. Trot and Cap'n Bill have adventures together in The Sea Fairies and Sky Island, the latter of which brings in Button-Bright and Polychrome from The Road to Oz. Once Baum returned to writing Oz books, it was perhaps inevitable that he would end up relocating Trot and the Cap'n to Oz, which he does in Scarecrow. He leaves a loose end, however, in that we never learn what happened to Trot's parents. Sure, her mother is kind of cranky and her father rarely around, but don't they at least deserve a note? Button-Bright, who comes to Oz in the company of Trot and Cap'n Bill, also had living parents the last time he had appeared. Glinda claims, "I do not see any way, at present, for them to return again to the outside world," despite the fact that Ozma had no trouble using the Magic Belt to send Dorothy back there in earlier books.
In some of Thompson's later books, she starts referring to Betsy and Trot as Princesses of Oz. Her explanation is that Ozma gave them royal titles to go along with Dorothy's, which makes sense, but then Dorothy is clearly Ozma's favorite. Trot would have already been ruler of the two main countries of Sky Island and a Princess of the Munchkins by this point, but I don't believe Betsy ever received a title in any other books.