I have an almost identical necklace except mine is a triple strand.They strike me as imitation pearls, but I could be wrong since real pearls will yellow if left in a box for years.
View attachment 466798However, they look a lot like the double strand of graduated Richelieu fake pearls that my grandmother gave me 50+ years ago (when I was 11). She told me they were "the best pearl that was not a pearl." They are plastic, and have yellowed over the years to the color your pearls are. They also have the same surface texture that your pearls have. Mine has a sterling silver clasp which is original to the necklace. Mine also has those end beads which are common on low end bead necklaces. I believe my grandmother bought these in the 1940s or 1950s. They could possibly go back to the late 1930s.
My grandmother treasured them as good fakes, so it's possible when your grandmother's pearls needed restringing, she made two strands so she could give each of you a strand of her pearls.
Richelieu started making fake pearls in 1911 before Mikimoto made cultured pearls popular. They were sold in satin lined boxes and presented like real pearls would be. They were most popular in the 1940s and 1950s. By the 1970s, the jewelry became more costume looking. The company was merged with several other companies over the decades, and by 2003, the Richelieu name was no longer used.