[erasmuse]

>do they also share common memories?

Indeed! The way I see it, identity derives from memory. So to the extent (if any) that they share identical memories they are strictly the same person. Beyond that I would guess there may be some (false) identification with behaviours of the other twin, in a manner redolent of Gazzaniga and Sperry's famous split brain experiment:

https://physics.weber.edu/carroll/honors/split_brain.htm

[mlevental]

redolent means to smell of. you mean reminiscent

[erasmuse]

Shouldn't you begin your sentences with capital letters? :-)