Your question draws an explicit link between celebrity and Damien Hirst. Yet when reflecting on the Young British Artists (YBAs), I think you could focus on how the other artists you mention also harness elements associated with celebrity or other forms of unwanted attention.
Celebrities are subject to constant scrutiny. Female celebrities, especially, are put under a microscope. One main element of celebrity culture, especially for women, sexual objectification. Their bodies are picked apart, their private accounts are hacked, and their relationships are weaponized against them.
The work of Sarah Lucas seems to address the sexual objectification that women — especially famous women — are subject to. Two Eggs and a Kebab addresses this theme. In it, Lucas arranged two eggs and a kebab (fresh each morning) on a table in a configuration that alludes to the physical features women are often reduced to—breasts and a vagina. A work like One Thousand Eggs: For Women, wherein she invited women to throw eggs against a gallery wall, seem to be a rejection of this objectification; it’s like she’s giving women a chance to fight back against all of the scrutiny that they receive and to work collaboratively to reclaim their agency. It’s also quite a provocative, public gesture. Of course, provocation and the concomitant publicity are not separate from celebrity culture—and this shock factor was pretty typical of the YBAs.
Tracey Emin’s My Bed also seems to speak to loss of privacy. Although the more obvious meaning of the piece is a commentary on grief, alcoholism, and depression, it also flips the script on the expectations for famous women and for female artists. As with a famous woman, her bed — or personal life — is on full display. It’s for everyone to see, and she has put it there. Furthermore, she has included objects that women are expected to keep private, such as period stained clothing, a pregnancy test, lubricant, and condoms. She is taking the scrutiny of her that is inevitable and using it to force focus on uncomfortable topics.
As for Damien Hirst, his celebrity and his work seems to have less to do with his personal identity. With Lucas, Emin, and Jenny Saville, they themselves feature heavily in their respective work. Yet Hirst himself is rather absent in his work. For example, The Physical Impossibility of Death in the Mind of Someone Living is a dead shark that has been suspended in a tank of formaldehyde. It forces viewers to confront an extremely scary and, as the title suggests, incomprehensible concept: the reality that they will one day die. Perhaps this lack of personal identity in his work speaks somewhat to the loss of personal identity that came with Hirst's success.