They known that the reason he's doing it is because he has to. If Clinton had maintained silence from the beginning instead of publicly slandering Lewinsky and other women as liars, then he might have some moral grounds to demand privacy. Moreover, Clinton is attempting to claim a right of privacy that he and his administration have actively denied to others. But the standard of consent is applied only to employment positions and not to mentor situations, such as an academic setting in which sex occurs between a professor and a student. In the latter, consent from the woman is irrelevant as are threats or promises of advancement from the man.
Sexual harassment is deemed to be present simply because of the power differential.
It is, by the definition now used as policy in virtually all American universities, an act of harassment and abuse for a man in a position of power to engage in sex with a woman, no matter how eager she may be. The President's relationship to a White House intern is clearly more of a mentor than an employer one. When the most powerful man on earth repeatedly entertains 'a certain kind of sex' from a young woman naive enough to believe they are having a 'relationship' thereby, the situation meets the standards of sexual harassment by a mentor. Clinton aggressively advocated exposing and punishing such powerful and abusive men through law and public policies. He is morally constrained from hiding behind privacy when such punishment comes knocking on his own door.